1.

0 INTRODUCTION

Matrix algebra is a fundamental component of modern mathematics, providing a

structured approach to solving complex problems across various disciplines. From
theoretical mathematics to practical applications in engineering, computer science,
physics, and economics, matrices serve as a powerful tool for analyzing linear systems,
performing transformations, and optimizing numerical computations. The ability to
represent and manipulate data in matrix form has led to significant advancements in
scientific computing, artificial intelligence, and big data analysis.

The study of matrices encompasses key mathematical operations such as matrix

addition, multiplication, inversion, and decomposition techniques, which are essential for
solving linear equations and performing eigenvalue analysis. Determinants, rank, and
trace provide deeper insights into matrix properties, influencing their stability and
computational efficiency. Moreover, special matrices like diagonal, symmetric,
orthogonal, and sparse matrices enable efficient computations and have widespread
applications in machine learning, signal processing, and control systems.

In computational mathematics, numerical algorithms such as Gaussian

elimination, LU decomposition, and Singular Value Decomposition (SVD) play a crucial
role in solving large-scale problems. These techniques help optimize performance in
high-dimensional data processing and are particularly useful in fields like graph theory,
quantum mechanics, financial modeling, and computer vision. The emergence of
specialized computing hardware and parallel processing has further enhanced the ability
to handle complex matrix operations efficiently. In recent years, advancements in
computational methods have enhanced the efficiency and applicability of matrix
operations. Techniques such as Gaussian elimination, LU decomposition, and singular
value decomposition (SVD) are widely used for solving large-scale numerical problems.

This study aims to provide a comprehensive exploration of matrix theory,

computational techniques, and real-world applications. It will focus on theoretical
foundations, algorithmic approaches, and their impact on various domains. By
understanding the structure and functionality of matrices, researchers and practitioners
can develop efficient solutions for data-intensive and computationally demanding tasks.

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2.0 OBJECTIVES

1. To understand the fundamental concepts of matrix algebra – Examining basic

operations such as matrix addition, multiplication, transposition, inversion, and
special properties like determinants, rank, and trace.

2. To analyze different types of matrices and their properties – Investigating

structured matrices such as diagonal, symmetric, orthogonal, sparse, and Toeplitz
matrices, and their computational advantages in numerical applications.

3. To study matrix transformations and their applications – Exploring concepts like

eigenvalues, eigenvectors, and singular value decomposition (SVD) to understand
their significance in fields such as data science, signal processing, and physics.

4. To evaluate numerical methods for solving matrix equations – Reviewing

computational techniques such as Gaussian elimination, LU decomposition, QR
decomposition, and iterative solvers for handling large-scale linear systems.

5. To explore the role of matrices in real-world applications – Investigating how

matrices are used in diverse fields, including machine learning, quantum mechanics,
image processing, cryptography, and control systems.

6. Comparing different algorithms for matrix factorization and optimization in high-

performance computing.

7. To investigate the importance of matrix algebra in artificial intelligence and big

data – Analyzing how matrices are applied in deep learning, recommendation
systems, and statistical modeling.

8. To develop a structured approach to implementing matrix operations in

programming – Demonstrating the use of computational tools such as MATLAB,
Python (NumPy), and R for efficient matrix manipulation and problem-solving.

9. To understand the limitations and challenges of matrix computations –

Examining issues such as numerical instability, computational complexity, and error
propagation in matrix-based calculations.

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3.0 LITERATURE REVIEW

The study of matrix algebra has evolved significantly over the years, with
extensive research focusing on its theoretical foundations, computational techniques, and
real-world applications. Matrices are widely used in fields such as engineering, physics,
computer science, artificial intelligence, and economics, making them a fundamental
topic in both pure and applied mathematics. This literature review explores the historical
development of matrix theory, key advancements in computational methods, and various
applications of matrices in modern technology and scientific research.

1. Historical Development of Matrix Theory

The concept of matrices dates back to the 19th century, with contributions from
mathematicians such as Arthur Cayley, James Joseph Sylvester, and Carl Friedrich
Gauss. Cayley introduced the notion of matrix multiplication and the characteristic
equation, which laid the foundation for modern linear algebra. Over time, matrix theory
expanded to include determinants, eigenvalues, eigenvectors, and matrix
decompositions, becoming a critical tool for solving linear equations and representing
linear transformations.

Early applications of matrices emerged in mechanics, electrical circuits, and statistics,

and their importance grew with the advent of digital computing in the 20th century. The
introduction of numerical algorithms for solving large-scale matrix problems
revolutionized scientific computing and led to advancements in fields such as finite
element analysis, control systems, and quantum mechanics.

2. Matrix Transformations and Decompositions

Matrix transformations play a fundamental role in linear algebra and applied

mathematics, allowing for efficient representation and computation of complex systems.
Several key transformations and decompositions have been extensively studied:

• Eigenvalue and Eigenvector Analysis – Eigenvalues and eigenvectors provide

insights into the behavior of linear transformations, particularly in stability
analysis, signal processing, and machine learning. Research has explored
various numerical methods for computing eigenvalues, such as the QR algorithm
and power iteration methods.

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 • Singular Value Decomposition (SVD) – SVD is widely used in image
compression, principal component analysis (PCA), and recommendation
systems. Studies have shown that low-rank approximations of matrices via SVD
can significantly enhance data processing efficiency.

• LU and QR Decomposition – These decompositions are essential for solving

linear systems and optimization problems, with LU decomposition frequently
applied in numerical simulations and QR decomposition used in least-squares
regression.

3. Computational Techniques for Matrix Algebra

With the rise of computational mathematics, numerous algorithms have been developed
to efficiently handle matrix operations. Research in numerical linear algebra has focused
on:

• Direct vs. Iterative Methods – Direct methods such as Gaussian elimination

and Cholesky decomposition provide exact solutions but may be
computationally expensive. Iterative methods like Jacobi, Gauss-Seidel, and
Krylov subspace methods are preferred for large, sparse matrices in high-
dimensional applications.

• Sparse Matrix Computations – Sparse matrices arise in areas such as graph

theory, network analysis, and scientific computing. Specialized algorithms like
compressed sparse row (CSR) format and Krylov subspace methods have
been developed to optimize memory usage and computational speed.

• Parallel and High-Performance Computing – Modern research explores

parallel processing and GPU acceleration for handling large-scale matrix
computations efficiently. Libraries such as BLAS (Basic Linear Algebra
Subprograms), LAPACK, and CUDA-based implementations have been
developed to enhance matrix operation speeds.

4. Applications of Matrices in Science and Engineering

Matrices are extensively used in various disciplines, with research highlighting their
significance in:

• Machine Learning and Artificial Intelligence – Matrices form the backbone of

deep learning, where large matrix multiplications are used in training neural

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 networks. Gradient descent optimization and backpropagation algorithms rely
on matrix differentiation and transformations.

• Quantum Mechanics and Physics – Quantum states and operations are

represented using Hermitian and unitary matrices, making matrix algebra a
fundamental tool in quantum computing and physics simulations.

• Signal Processing and Image Compression – Techniques like Fourier

transforms, wavelet transforms, and JPEG compression heavily rely on
matrix operations. SVD-based compression methods have been shown to optimize
storage without significant loss of quality.

• Economics and Finance – In finance, matrices are used for portfolio

optimization, risk analysis, and econometric modelling. The Markowitz
portfolio theory employs covariance matrices to balance risk and return in
investment strategies.

• Control Systems and Robotics – Matrices are fundamental in state-space

representation, helping engineers design control algorithms for autonomous
systems, robotic kinematics, and stability analysis.

5. Challenges and Future Research Directions

Despite extensive research on matrix algebra, several challenges remain:

• Computational Complexity – As matrix sizes increase, computational costs

become a major concern. Future research is focused on developing more efficient
algorithms for large-scale and high-dimensional matrix operations.

• Numerical Stability and Precision – Many matrix computations suffer from

numerical instability, leading to rounding errors. Researchers are working on
improved conditioning techniques and adaptive precision algorithms.

• Integration with Emerging Technologies – With the rise of quantum

computing, AI-driven optimizations, and cloud computing, matrix operations
are evolving to handle large datasets and real-time computations more
effectively.

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CHAPTER I: INTRODUCTION OF MATRICES

1.1 Definition 1:
A rectangular arrangement of mn numbers, in m rows and n columns and enclosed
within a bracket is called a matrix. We shall denote matrices by capital letters as A, B, C
etc.

A is a matrix of order m n. ith row jth column element of the matrix denoted by
Remark: A matrix is not just a collection of elements but every element has assigned a
definite position in a particular row and column.
1.2 Special Types of Matrices:
1. Square matrix:
A matrix in which numbers of rows are equal to number of columns is
called a square matrix.
Example:

2. Diagonal matrix:

A square matrix A = is called a diagonal matrix if each of its non-diagonal

element is zero.

That is and at least one element .

Example:

3. Identity Matrix
A diagonal matrix whose diagonal elements are equal to 1 is called identity
matrix and denoted by .

That is
Example:

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4. Upper Triangular matrix:

A square matrix said to be a Upper triangular matrix if .

Example:

5. Lower Triangular Matrix:

A square matrix said to be a Lower triangular matrix if .

Example:

6. Symmetric Matrix:

A square matrix A = said to be a symmetric if for all i and j.

Example:

7. Skew- Symmetric Matrix:

A square matrix A = said to be a skew-symmetric if for all i

and j.
Example:

8. Zero Matrix:
A matrix whose all elements are zero is called as Zero Matrix and order

Zero matrix denoted by .

Example:

9. Row Vector
A matrix consists a single row is called as a row vector
or row matrix.

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Example:

10. Column Vector

A matrix consists a single column is called a column vector or column matrix.
Example:

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CHAPTER 2:

MATRIX ALGEBRA

2.1. Equality of two matrices:

Two matrices A and B are said to be equal if
(i) They are of same order.
(ii) Their corresponding elements are equal.
then
That is if A = for
all i and j.
2.2. Scalar multiple of a matrix

Let k be a scalar then scalar product of matrix A = given denoted

by kA and given by kA = or

2.3. Addition of two matrices:

and
Let A = are two matrices with same order then sum of
the two matrices are given by

Example 2.1: let

and

Example 2.2: Let

Calculate (i) AB
(ii) BA
(iii) is AB = BA ?
2.5. Integral power of Matrices:
Let A be a square matrix of order n, and m be positive integer then we
define
(m times multiplication)
2.6. Properties of the Matrices
Let A, B and C are three matrices and are scalars then

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 (i) Associative Law
(ii) Distributive law
(iii) Associative Law
(iv) Associative Law
(v) Associative Law
(vi) Distributive law
2.7. Transpose:

The transpose of matrix A = , written ( is the matrix

obtained by writing the rows of A in order as columns.

That is .
Properties of Transpose:
(i)
=A
(ii)
=k for scalar k.
(iii)
(iv)
Example 2.3: Using the following matrices A and B, Verify the transpose properties

Proof: (i) Let and are the element of the matrix A and B
respectively. Then is the element of matrix and it is
element of the matrix

Also and are the element of the matrix and respectively.

Therefore is the element of the matrix +
(ii) Let element of the matrix A is , it is element of the then it is
element of the matrix
(iii) try

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 is the
(iv) element of
the AB It is result of the multiplication of the ith row and kth column and it
is element of the matrix .
lement is the multiplication of kth row of with ith column of , That is kth column of
B with ith row of A.
A square matrix A is said to be symmetric if .
Example:

, A is symmetric by the definition of

symmetric matrix.
Then

That is
A square matrix A is said to be skew- symmetric if
Example:

(i) and are both symmetric.

(ii) is a symmetric matrix.
(iii) is a skew-symmetric matrix.
(iv) If A is a symmetric matrix and m is any positive integer
then is also symmetric.
(v) If A is skew symmetric matrix then odd integral powers
of A is skew symmetric, while positive even integral
powers of A is symmetric.
If A and B are symmetric matrices then
(vi) is symmetric.
(vii) is skew-symmetric.
Exercise 2.1: Verify the (i) , (ii) and (iii) using the following matrix A.

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CHAPTER 3:

DETERMINANT, MINOR AND ADJOINT MATRICES

Definition 3.1:

Let A = be a square matrix of order n , then the number called

determinant of the matrix A.
(i) Determinant of 2 2 matrix
then = =
Let A=
(ii) Determinant of 3 3 matrix

Let B =
=
Then

Exercise 3.1: Calculate the determinants of the following matrices

(i) (ii)
3.1 Properties of the Determinant:
a. The determinant of a matrix A and its transpose are equal.

b. Let A be a square matrix

(i) If A has a row (column) of zeros then
(ii) If A has two identical rows ( or columns) then
c. If A is triangular matrix, then is product of the diagonal elements.
d. If A is a square matrix of order n and k is a scalar then
3.2 Singular Matrix
If A is square matrix of order n, the A is called singular matrix when
and non- singular otherwise.
3.3. Minor and Cofactors:

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 Let A = is a square matrix. Then denote a sub matrix of A
with order (n-1)

(n-1) obtained by deleting its row and column. The determinant is

called the minor of the element of A.

The cofactor of denoted by and is equal to .

Exercise 3.2: Let

(i) Compute determinant of A.
(ii) Find the cofactor matrix.
3.4. Adjoin Matrix:

The transpose of the matrix of cofactors of the element of A denoted by is

called adjoin of matrix A.

Example 3.3: Find the adjoin matrix of the above example.

Theorem 3.1:
For any square matrix A,
where I is the identity matrix of same order.

Proof: Let A =
Since A is a square matrix of order n, then also in same order.
Consider

then

Now consider the product

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 and
( as we know that when )

Where is unit matrix of order n.

Theorem 3.2: If A is a non-singular matrix of order n, then
.
Proof: By the theorem 1

Theorem 3.3: If A and B are two square matrices of order n then

Proof: By the theorem 1

Therefore
Consider ,

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 ……………… (i)
Also consider

…………….. (ii)
Therefore from (i) and (ii) we conclude that

Some results of adjoint

(i) For any square matrix A
(ii) The adjoint of an identity matrix is the identity matrix.
(iii) The adjoint of a symmetric matrix is a symmetric matrix.

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 CHAPTER 4:

INVERSE OF A MATRIX AND ELEMENTARY ROW

OPERATIONS

4.1 Inverse of a Matrix

Definition 4.1:
If A and B are two matrices such that , then each is said to
.
be inverse of the other. The inverse of A is denoted by
Theorem 4.1: (Existence of the Inverse)
The necessary and sufficient condition for a square matrix A to have an inverse is that

(That is A is non singular).

Proof: (i) The necessary condition
Let A be a square matrix of order n and B is inverse of it, then

Therefore .
(ii) The sufficient condition:
If , the we define the matrix B such that

Then

Similarly =
Thus hence B is inverse of A and is given by

Theorem 4.2: (Uniqueness of the Inverse)

Inverse of a matrix if it exists is unique.
Proof: Let B and C are inverse s of the matrix A then
and

17
 find
Example 6: Let
Theorem 4.3: (Reversal law of the inverse of product)
If A and B are two non-singular matrices of order n, then (AB) is also non singular
and
.
Proof:
Since A and B are non-singular , therefore
, then .
Consider
…………(1)
Similarly
…………..(2)
From (1) and (2)
=

Therefore by the definition and uniqueness of the inverse

If are non singular Corollary4.1: matrices
of order n,
then
.
Theorem 4.4: If A is a non-singular matrix of order n then .
Proof: Since therefore the matrix is non-singular and exists.
Let
Taking transpose on both sides we get

Therefore
That is .

Theorem 4.5: If A is a non-singular matrix , k is non zero scalar, then .

Proof: Since A is non-singular matrix exits.

Let consider
Therefore is inverse of

18
 By uniqueness if inverse
Theorem 4.6: If A is a non-singular matrix then

.
Proof: Since A is non-singular matrix, exits and we have

Therefore

Then
4.2 Elementary Transformations:
Some operations on matrices called as elementary transformations. There are six types
of elementary transformations, three of then are row transformations and
other three of them are column transformations. There are as follows
(i) Interchange of any two rows or columns.
(ii) Multiplication of the elements of any row (or column) by a non
zero number k.
(iii) Multiplication to elements of any row or column by a scalar k and
addition of it to the corresponding elements of any other row or
column.
We adopt the following notations for above transformations
(i) Interchange of ith row and jth row is denoted
.
by .
(ii) Multiplication by k to all elements in the ith row
(iii) Multiplication to elements of jth row by k and adding them to the

corresponding elements of ith row is denoted by .

4.2.1 Equivalent Matrix:
A matrix B is said to be equivalent to a matrix A if B can be obtained from A, by for
forming finitely many successive elementary transformations on a matrix A.
Denoted by A~ B.

4.3 Rank of a Matrix:

Definition:
A positive integer ‘r’ is said to be the rank of a non- zero matrix A if

19
 (i) There exists at least one non-zero minor of order
r of A and (ii) Every minor of order greater than
r of A is zero.
The rank of a matrix A is denoted by .
4.4 Echelon Matrices:
Definition 4.3:

A matrix is said to be echelon form (echelon matrix) if the

number of zeros preceding the first non zero entry of a row increasing by row
until zero rows remain.
In particular, an echelon matrix is called a row reduced echelon matrix if the
distinguished elements are
(i) The only non- zero elements in their respective
columns. (ii) Each equal to 1.
Remark: The rank of a matrix in echelon form is equal to the number of non-zero rows
of the matrix.
Example 4.1:
Reduce following matrices to row reduce echelon
form
(i)
(ii)

20
CHAPTER 5

SOME APPLICATIONS OF MATRICES AND DETERMINANTS

5.0. Introduction
A few applications in solving difference and differential equations, applications in
evaluating Jacobians of matrix transformations, optimization problems, probability
measures and Markov processes and some topics in statistics will be discussed in this
chapter.

5.1. Difference and Differential Equations

In order to introduce the idea of how eigenvalues can be used to solve difference and
differential equations a few illustrative examples will be done here.

5.1.1. Fibonacci sequence and difference equations The famous Fibonacci sequence is
the following:
0,1,1,2,3,5,8,13,21,...
where the sum of two consecutive numbers is the next number. Surprisingly, this sequence
appears in very many places in nature. Consider a living micro organism such as a cell
which is reproducing in the following fashion: To start with there is one mother. The
mother cell needs only one unit of time to reproduce. Each mother produces only one
daughter cell. The daughter cell needs one unit of time to grow and then one unit of time
to reproduce. Let us examine
374
the population size at each stage.

stage1 number = 1 one mother at the first unit of time

stage 2 number = 1 1 mother only

stage 3 number = 2 1 mother +1 young daughter

stage 4 number = 3 1 mother, 1 mature and 1 young daughters

stage 5 number = 5 2 mothers, 1 mature and 2 young daughters

21
and so on. The population size follows the sequence 1,1,2,3,5,8,... the famous Fibonacci
sequence.
If you look at the capitulum of a sunflower the florets, or the seeds when the florets
become seeds, seem to be arranged along spirals starting from the periphery and going
inward. You will see one set of such radial spirals going in one direction and another set
of radial spirals going in the opposite direction. These numbers are always two successive
numbers from a Fibonacci sequence. In a small sunflower it may be (3,5), in a slightly
larger flower it may be (5,8) an so on. Arrangement of florets on a pinapple, thorns on
certain cactus head, leaves on certain palm trees, petals in dhalias and in very many such
divergent places one meets Fibonacci sequence. A theory of growth and forms,
explanation for the emergence of Fibonacci sequence and a mathematically reconstructed
sunflower head and many other details can be seen from the paper, Mathai, A.M. and
Davis, T.A. [1974, Mathematical Biosciences, Vol.20, pp.117–133]. Incidently the
journal, Mathematical Biosciences, has adapted the above mathematically reconstructed
sunflower model as its cover design from 1976 onward.
If the Fibonacci number at the k-th stage is denoted by Fk then the number at the (k + 2)-
th stage is
Fk+2 = Fk+1 + Fk.
(5.1.1)
This is a difference equation of order 2. Fk+1 − Fk is a first order difference and Fk+2 −Fk+1
is again a first order difference. Then going from Fk to Fk+2 is a second order difference.
That is, (5.1.1) is a second order difference equation. One way of computing Fk for any k,
k may be 10385, is to go through properties of matrices. In order to write a matrix equation
let us introduce dummy equations such as Fk = Fk,
Fk+1 = Fk+1 and so on. Consider the equations
Fk+2 = Fk+1 + Fk
Fk+1 = Fk+1
(5.1.2)

22
and

.
Then the two equations in (5.1.2) can be written as

.
(5.1.3)

Let us assume F0 = 0 and F1 = 1 which means . Then from (5.1.3) we have

V1 = AV0, V2 = AV1 = A(AV0) = A2V0,...,Vk = AkV0.
In order to compute Vk we need to compute only Ak since V0 is known. Straight
multiplication of A with A for a total of 10385 times is not an easy process. We will use
the property that the eigenvalues of Ak are the k-th powers of the eigenvalues of A, sharing
the same eigenvectors of A. Let us compute the eigenvalues and the eigenvectors of A.

An eigenvector corresponding to is given by

, or

is an eigenvector. Similarly an eigenvector corresponding to is given by

.
Let

23
Therefore

.
Since A and Ak share the same eigenvectors we have

.
Hence

.
Therefore

Since will approach zero when k is large.

.
(5.1.4)

Evidently, one has to take only powers of λ1 when k is large. That is, for large

k. This number is known as the “golden ratio” which appears in nature at many
places.
One general observation that can be made is that we have an equation of the type
Vk = AkV0 = QΛkQ−1V0
for an n×n matrix A where Λ = diag(λ1,...,λn) and Q is the matrix of eigenvectors of A,
assuming |Q| 6= 0. Then setting Q−1V0 = C, C0 = (c1,...,cn) we have
.
If X1,...,Xn are the eigenvectors of A, constituting Q = (X1,...,Xn), then
λk1c1 Vk = Q(ΛkQ−1V0) = (X1,...,Xn) ...

24
λkncn
= c1λk1X1 + ... + cnλknXn
which is a linear function of or a linear combination of the so called
pure solutions λki Xi.
Example 5.1.1. Suppose that a system is growing in the following fashion. The first stage
size plus 3 times the second stage size plus the third stage size is the fourth stage size. Let
F0 = 0, F1 = 1, F2 = 1 be the initial conditions. Then
F3 = 0 + 3(1) + 1 = 4, F4 = 1
+ 3(1) + 4 = 7,
F5 = 1 + 3(4) + 7 = 19,
and so on. Then for any k we have
Fk + 3Fk+1 + Fk+2 = Fk+3.
Compute Fk for k = 100.
Solution 5.1.1. Consider the following set of equations:
Fk + 3Fk+1 + Fk+2 = Fk+3
Fk+2 = Fk+2 Fk+1 = Fk+1 ⇒

Let and . Then we have

AUk = Uk+1 and ;

AU0 = U1, U2 = A2U0,...
Then
Uk = AkU0.
Let us compute the eigenvalues of A. Consider the equation

.
Obviously λ = −1 is one root. Dividing√ −λ3+λ2+3λ+1 by λ√+1 we have −√λ2+2λ+1.

The other two roots are [1 ± 2]. Then λ1 = −1, λ2 = 1 + 2, λ3 = 1 − 2 are the roots. Let us
compute some eigenvectors corresponding to these roots. For λ = −1

25
is one vector. For

√
is one vector. For λ3 = 1 − 2

is one vector. Let

,
Therefore

.
But

and

26
√
When k → ∞ we have (1− 2)k → 0. Thus for k large a good approximation to Uk is the
following:

.
Hence for
,
5.1.2. Population
growth
Consider, for example competing populations of foxes and rabbits in a given region. If
there is no rabbit available to eat the foxes die out. If rabbits are available then for every
kill the population of foxes has a chance of increasing. Suppose that the observations are
made at the end of every six months, call them stages 0,1,2,... where stage 0 means the
starting number. Let Fi and Ri denote the fox and rabbit populations at stage i. Suppose
that the growth of fox population is governed by the difference equation
Fi+1 = 0.6Fi + 0.2Ri.
Left alone the rabbits multiply. Thus the rabbit population is influenced by the natural
growth minus the ones killed by the foxes. Suppose that the rabbit population is given by
the equation
Ri+1 = 1.5Ri − pFi
where p is some number. We will look at the problem for various values of p. Suppose
that the initial populations of foxes and rabbits are 10 and 100 respectively. Let us denote
by

.
Then the above difference equations can be written as

or

27
 .
Thus
X1 = AX0, X2 = AX1 = A2X0,...,Xk = AkX0.
For example, at the first observation period the population sizes are given by

.
For example, for p = 1, the numbers are
F1 = (0.6)(10) + (0.2)(100) = 26
and
R1 = −1(10) + 1.5(100) = 140.
Let us see what happens in the second stage with the same p, that is for k = 2, p = 1. This
can be computed either from the first stage values and by using A or from the initial values
and by using A2. That is,

.
Note that for p = 1 the fox population and the rabbit population will explode eventually.
Let us see what happens if p = 5. Then

.
Note that at the next stage the rabbits will disappear and from then on the fox population
will start decreasing at each stage.
Growths of interdependent species of animals, insects, plants and so on are governed by
difference equations of the above types. If there are three competing populations involved
then the coefficient matrix A will be 3 × 3 and if there are n such populations then A will
be n×n, n ≥ 1. The long-term behavior of the populations can be studied by looking at the
eigenvalues of A because when A is representable as
A = QDQ−1, D = diag(λ1,...,λn) ⇒ Ak = QDkQ−1
where λ1,...,λn are the eigenvalues of the n × n matrix A. Then λk → 0 as k → ∞ when |λ| <
1 and λk → ∞ for λ > 1 as k → ∞. Thus the eventual extinction or explosion or stability of
the populations is decided by the eigenvalues of A.

28
5.1.3. Differential equations and their solutions
Consider a system of total differential equations of the linear homogeneous type with
constant coefficients. Suppose that a supermarket has barrels of almonds and pecans (two
competing types of nuts as far as demand is concerned). Let u denote the amount of stock,
in kilograms (kg) of almonds and v that of pecans. The store fills up the barrels according
to the sales. The store finds that the rate of change of u over time is a linear function of u
and v, so also the rate of change of v over time t. Suppose that the following are the
equations.
d u
=2 u + v
dt
v
d = u +2 v

dt
which means
At the start of the observations, t = 0, suppose that the stock is u = 500kg and

v = 200kg. If W is the vector then we say that the initial value of W, denoted by

. We want to solve (5.1.5) with this initial value. The

differential equations in (5.1.5) are linear and homogeneous in u and v with u and v having
constant (free of t) coefficients. The method that we will describe here will work for n
equations in n variables u1,...,un, where each is a function of another independent variable
such as t, ui = ui(t), i = 1,...,n, and when the right sides are linear homogeneous with
constant coefficients. For simplicity we consider only a two variables case.
If there was only one equation in one variable of the type in (5.1.5) then the equation
would be of the form
du
= au
dt
where a is a known number. Then the solution is
u = eatu0 if u = u0 at t = 0(initial value).
(5.1.6)
Then in the case of two equations as in (5.1.5) we can search for solutions of the type in
(5.1.6). Let us assume that
u = eλtx1 and v = eλtx2,

29
 (5.1.7)
for some unknown λ, the same λ for both u and v, x1 and x2 are some parameters free of t.
Substituting these in (5.1.5) we obtain
λeλtx1 = 2eλtx1 + eλtx2
λeλtx2 = eλtx1 + 2eλtx2.Canceling eλt and writing the equations in matrix form we(5.1.8)
have

.
(5.1.9)
The problem reduces to that of finding the eigenvalues and eigenvectors of A. The
eigenvalues are given by

An eigenvector corresponding to λ = 1 is given by

is one vector. Corresponding to λ2 = 3,

is one vector. For λ = λ1 = 1 a solution for W is

.
(5.1.10)
For λ = λ2 = 3 a solution for W is

.
(5.1.11)
Any linear function of W1 and W2 is again a solution for W. Hence a general solution for
W is

30
 (5.1.12)
where c1 and c2 are arbitrary constants. Let us try to choose c1 and c2 to satisfy the

initial condition, for t = 0. Letting t = 0 in (5.1.12) we have

Then the solution to the equation in (5.1.5) is

.
Since the exponents are positive, ebt → ∞ as t → ∞ when b > 0, u and v both increase with
time. In fact, the eigenvalues λ1 = 1 and λ2 = 3, appearing in the exponents, measure the
rate of growth. This can be noticed from the pure solutions in (5.1.10) and (5.1.11). A
mixture of these pure solutions is what is given in (5.1.12). If an eigenvalue λ is positive,
as in (5.1.10) and (5.1.11), then eλt → ∞ as t → ∞. In this case we say that the equations
are unstable. If λ = 0 the equations are said to be neutrally stable. When λ < 0,eλt → 0 as
t → ∞. In this case we say that the equations are stable. In our example above, the pure
solutions for both λ1 = 1 and λ2 = 3, as seen from (5.1.10) and (5.1.11), are unstable.
A slightly more general situation arises if there are some constant coefficients for
du dv dt and dt in (5.1.5).
Example 5.1.2. Solve the following system of differential equations if u and v are
functions of t and when t = 0,u = 100 = u0 and v = 200 = v0:
du
2= 2u + v dt
dv
3= u + 2v. (5.1.13) dt
Solution 5.1.2. Divide the first equation by 2 and the second equation by 3. Then the
problem reduces to that in (5.1.5). But if we want to avoid fractions at the beginning stage
itself of solving the system, or to solve the system as they are in (5.1.13), then we look
for a solution of the type
u = eλtx1, v = eλtx2

31
for some λ and for some constants x1 and x2. [Observe that if the original system of
equations has some fractional coefficients then multiply the system by appropriate
numbers to make the coefficients non-fractional. Then the following procedure can be
applied.] Then the equations in (5.1.13) reduce to the following form:
2λeλtx1 = 2eλtx1 + eλtx2
3λeλtx2 = eλtx1 + 2eλtx2. Canceling eλt and writing

we have,

If this equation has a non-null solution then

Let us compute X corresponding to λ1 and λ2. For

.
One solution for X is

Similarly for one solution is

.
For λ = λ1 one solution for W is

and for λ = λ2 the solution for W is

.
Thus a general solution for W is W = c1W1 + c2W2 where c1 and c2 are arbitrary constants.
That is,

32
and

.
But for t = 0, u = u0 = 100 and for t = 0,v = v0 = 200. That is,

.
Solving for c1 and c2 we have
√ √
c1 = 50(1 + 7) and c2 =
50(1 − 7).
Hence the general solution is,

Note that the same procedure works if we have m-th order equations of the type
dt
where b1,...,bk and aij’s are all constants and uj, j = 1,...,k are functions of t. In this case
look for a solution of the type uj = eµtxj, j = 1,...,k with the same µ and xj’s are some
quantities free of t. Then the left sides of (5.1.14) will contain µm. Put λ = µm. Then the
problem reduces to the one in Example 5.1.2.
Higher order differential equations can also be solved by using the same technique as
above. In order to illustrate the procedure we will do a simple example here.
Example 5.1.3. Let y be a function of t and let y0,y00,y000 denote the first order, second
order and third order derivatives respectively. Solve the following differential equation
by using eigenvalue method.
y000 − 4y00 + 3y0 = 0.
Solution 5.1.3. The classical way of doing the problem is to search for an exponential
solution of the type y = eλt. Then we get the characteristic equation
λ3 − 4λ2 + 3λ = 0 ⇒ λ1 = 0, λ2 = 1, λ3 = 3

33
are the solutions of this characteristic equation. Hence the three pure exponential solutions
are e . Now let us do the same problem by using eigenvalues.
Let
u = y0, v = y00 = u0, v0 = 4v − 3u and
y y0 y0

W= u ⇒W0= u0 = y00 .v v0 y000

Writing the above three equations in terms of
the vector W and its first derivative we
have
d
0 −3 4 v

. (5.1.15)
Now, compare with (5.1.5). We have a first order system in W. Let y = eλtx1, u = eλtx2, v =
eλtx3 for some x1,x2,x3 free of t. Then substituting in (5.1.15) and canceling eλt the equation
W 0 = AW reduces to the form

(5.1.16)
or the problem reduces to an eigenvalue problem. The eigenvalues of A are λ1 = 0, λ2 = 1,
λ3 = 3. Some eigenvectors corresponding to these eigenvalues are the following: which
gives
1 et
W1 = eλ1tX1 = 0 , W2 = eλ2tX2 = et ,
0 et
e3t
W3 = eλ3tX3 = 3e3t .
9e3t
Thus the pure solutions for y are 1,et and e3t. A general solution for y is then
y = c1 + c2et + c3e3t
(5.1.17)
where c1,c2,c3 are arbitrary constants.

34
CHAPTER 6

MATRIX SERIES AND ADDITIONAL PROPERTIES OF MATRICES

6.0. Introduction
The ideas of sequences, polynomials, series, convergence and so on in scalar variables
will be generalized to matrix variables in this chapter. We start with some basic properties
of polynomials and then see what happens if the scalar variable in the polynomial is
replaced by a square matrix.
6.1. Matrix Polynomials
Here a “matrix polynomial” does not mean a matrix where the elements are polynomials
in a scalar variable such as

.
Such a matrix will be called a matrix of polynomials. The term “matrix polynomial” will
be reserved for the situation where to start with we have a polynomial in a scalar variable
and we are replacing the scalar variable by a square matrix to obtain a polynomial in a
square matrix. For example, consider a polynomial of degree m in the scalar variable x,
p(x) = a0 + a1x + ... + amxm, am 6= 0
(6.1.1)
where a0,...,am are known constants. For example,
p1(x) = 4 + 2x − 3x2,a polynomial in x of degree 2; p2(x) = 2 + 5x,
a polynomial in x of degree 1; p3(x) = 7,
a polynomial in x of degree 0.
478
Let

.
Let us try to construct polynomials p1(A),p2(A),p3(A) in the matrix A, corresponding to
the scalar polynomials p1(x),p2(x),p3(x) above. When x in (6.1.1) is replaced by the matrix
A then the constant term a0 will be replaced by a0I, I the identity
which is again a 3 × 3 matrix. The following results are obviously true for matrix
polynomials.

35
We can note that the factorization properties also go through.

Consider the characteristic polynomial of an n × n matrix A. That is,

p(λ) = |A − λI| = (λ1 − λ)(λ2 − λ)...(λn − λ)
where λ1,...,λn are the eigenvalues of A and p(λ) = 0 is the characteristic equation. Then it
is easy to see that p(A) = O. That is,

6.1.1. Lagrange interpolating polynomial

Consider the following polynomial where λ1,...,λn are distinct quantities free of λ and
a1,...,an are constants:

which is a polynomial of degree n − 1 in λ. Put λ = λ1 in (6.1.3). Then we have p(λ1) = a1.

Similarly p(λj) = aj, j = 1,...,n. Therefore

.
(6.1.4)
The polynomial in (6.1.4) is called Lagrange interpolating polynomial. A more general
polynomial in this category, allowing multiplicities for λ1,...,λn is Hermite interpolating
polynomial which we will not discuss here. From (6.1.4) we have, for any square matrix
A, and p(λ) satisfying (6.1.4),

.
(6.1.5)
An interesting application of (6.1.5) is that if λ1,...,λn are the distinct eigenvalues of any
n×n matrix A and p(λ) is any polynomial of the type in (6.1.4) then the matrix p(A) has
the representation in (6.1.5). Let us do an example to highlight this point. Example 6.1.1.
Compute e5A where

36
 .
Solution 6.1.1. The eigenvalues of A are obviously λ1 = 1, λ2 = 4. Let p(λ) = e5λ.
Then from (6.1.4)

Therefore from (6.1.5)

6.1.2. A spectral decomposition of a matrix

We will consider the spectral decomposition of a matrix A when the eigenvalues are
distinct. The results hold when some of the eigenvalues are repeated also. In the repeated
case we will need Hermite interpolating polynomials to establish the results. When the
eigenvalues of A are distinct we have the representation in (6.1.5) where p(λ) is a
polynomial defined on the set of distinct eigenvalues of A (spectrum of A). Let (6.1.5) be
written as
p(A) = A1 + ... + An .
(6.1.6)
Let us consider the product A1A2. Excluding the constant parts, A1 and A2 are given by
A1 → (A − λ2I)(A − λ3I)...(A − λnI)
and
A2 → (A − λ1I)(A − λ3I)...(A − λnI).
Then
A1A2 → (A − λ1I)(A − λ2I)(A − λ3I)2...(A − λnI)2.
But from property (iii),
(λ1I − A)(λ2I − A)...(λnI − A) = O
and hence A1A2 = O. Similarly AiAj = O for all i and j, i =6 j. Thus A1,...,An are mutually
orthogonal matrices and hence linearly independent. Taking p(λ) = 1 in (6.1.6) we have
the relation

37
I = B1 + ... + Bn
(6.1.7)
where

and
BiBj = O for all i 6= j.
Then multiply both sides of (6.1.7) by Bj we have Bj = Bj2 for each j, j = 1,...,n.
Taking p(λ) = λ or p(A) = A in (6.1.6) we have the following spectral decomposition for
A:

This can be observed from property (iii) and (6.1.7). Note that
ABj = (A − λjI + λjI)Bj = (A − λjI)Bj + λjBj = λjBj
since (A − λjI)Bj = O by property (iii). Hence
λ1B1 + ... + λnBn = A(B1 + ... + Bn) = A
since B1+...+Bn = I by (6.1.7). We can also notice some more interesting properties from
(6.1.8).
BiBj = O = BjBi, i 6= j
as well as
BjA = ABj = λjBj2 = λjBj.
Thus the matrices A,B1,...,Bn commute and hence all can be reduced to diagonal forms by
a nonsingular matrix Q such that
D = λ1D1 + ... + λnDn
(6.1.9)
where QAQ−1 = D, QBjQ−1 = Dj for all j, DiDj = O for all i 6= j. The matrices B1,...,Bn in
(6.1.8) are also called the idempotents of A, different from idempotent matrices.

Example 6.1.2. For the matrix verify (6.1.7) and (6.1.8).

Solution 6.1.2. The eigenvalues are λ1 = 4 and λ2 = −1. Two eigenvectors corresponding
to λ1 and λ2 are

.
Let

38
 ;

Example 6.1.3. For the matrix A in Example 6.1.2 compute Q such that Q−1AQ =
diagonal. Also establish (6.1.9).
Solution 6.1.3. By straight multiplication
and
.
Taking the linear combination (6.1.9) is established.
6.1.3. An application in statistics
In the spectral decomposition of an n×n matrix A, as given in (6.1.8), each Bj is
idempotent. If A is real symmetric then Bj, j = 1,...,n are also real symmetric since the
eigenvalues of a real symmetric matrix are real. If the eigenvalues of A are all distinct
then each Bj is of rank 1. Consider X an n×1 real Gaussian vector random variable having
a standard Gaussian distribution. In our notation X ∼ Nn(O,I) where I is an identity matrix.
Consider the quadratic form X0AX. Then
X0AX = λ1X0B1X + ... + λnX0BnX.
Since B1 = Bj0 = Bj2 and since X ∼ Nn(O,I) it follows that , that is, X0BjX is a
real chisquare random variable with one degree of freedom. Since BiBj = O, i 6= j these
chisquare random variables are mutually independently distributed. Thus one has a
representation
X0AX = λ1y1 + ... + λnyn

39
where the y1,...,yn are mutually independently distributed chisquare random variables with
one degree of freedom each when the λj’s are distinct. One interesting aspect is that in
each Bj all the eigenvalues of A are present.
Exercises 6.1
6.1.1. If A is symmetrically partitioned to the form

then show that for any positive integer n,

where
6.1.2. Compute e−2A where

.
6.1.3. Compute sinA where

.
6.1.4. Spectrum of a matrix A. The spectrum of a matrix is the set of all distinct
eigenvalues of A. If B = QAQ−1 and if f(λ) is a polynomial defined on the spectrum of A
then show that
f(B) = Qf(A)Q−1.
Prove the result when the eigenvalues are distinct. The result is also true when some
eigenvalues are repeated.
6.1.5. If A is a block diagonal matrix, A = diag(A1,A2,...,Ak), and if f(λ) is a polynomial
defined on the spectrum of A then show that
f(A) = diag(f(A1),f(A2),...,f(Ak)).
6.1.6. If λ1,...,λn are the eigenvalues of an n×n matrix A and if f(λ) is a polynomial defined
on the spectrum of A then show that the eigenvalues of f(A) are f(λ1), f(λ2),...,f(λn).
6.1.7. For any square matrix A show that ekA, where k is a nonzero scalar, is a nonsingular
matrix.

40
6.1.8. If A is a real symmetric positive definite matrix then show that there exists a unique
Hermitian matrix B such that A = eB.
6.1.9. By using the ideas from Exercise 6.1.3, or otherwise, show that for any n×n matrix

6.1.10. For the matrix compute lnA, if it exists.

6.2. Matrix Sequences and Matrix Series
We will introduce matrix sequences and matrix series and concepts analogous to
convergence of series in scalar variables. A few properties of matrix sequences will be
considered first. Then we will look at convergence of matrix series and we will also
introduce a concept called “norm of a matrix”, analogous to the concept of “distance” in
scalar variables, for measuring rate of convergence of a matrix series.
6.2.1. Matrix sequences
Let A1,A2,... be a sequence of m × n matrices so that the k-th member in this sequence of

matrices is Ak. Let the (i,j)-th element in Ak be denoted by so that

. The elements are real or complex numbers.

Definition 6.2.1. Convergence of a sequence of matrices. For scalar sequences we say

that the limit of , as k → ∞, is aij if there exists a finite number aij such that
when k → ∞. Convergence of a matrix sequence is defined through element-wise

convergence. Thus if for all i and j when k → ∞ we say that Ak converges to A

= (aij) as k → ∞.
Example 6.2.1. Check for the convergence of the sequence A1,A2,... as well as that of the
sequence B1,B2,... where

.
Solution 6.2.1. Let us check the sequence A1,A2,.... Here

.
Hence

41
and the sequence is a convergent sequence. Now, consider B1,B2,.... Here

.
Evidently

.
But (−1)k oscillates from −1 to 1 and hence there is no limit as k → ∞. Also ek → ∞ when
k → ∞. Hence the sequence B1,B2,... is divergent.

The following properties are evident from the definition itself.

By combining with the ideas of matrix polynomials from Section 6.1 we can establish the
following properties: Since we have only considered Lagrange interpolating polynomials
in Section 6.1 we will state the results when the eigenvalues of the n × n matrix A are
distinct. But analogous results are available when some of the eigenvalues are repeated
also.
(iii) Let the scalar functions f1(λ),f2(λ),... be defined on the spectrum of an n × n matrix
A and let the sequence A1,A2,... be defined as Ak = fk(A), k = 1,2,.... Then the sequence
A1,A2,... converges, for k → ∞, if and only if the scalar sequences {f1(λ1),f2(λ1),...},
{f1(λ2),f2(λ2),...},...,{f1(λn),f2(λn),...} converge, as k → ∞, where λ1,..., λn are the
eigenvalues of A.

42
Example 6.2.2. For the matrix A show that

e , where .
√
Solution 6.2.2. The eigenvalues of A are ±i, i = −1. Take p(λ) = eλt and apply (6.1.5) of
Section 6.1. Then

.
6.2.2. Matrix series
A matrix series is obtained by adding up the matrices in a matrix sequence. For example
if A0,A1,A2,... is a matrix sequence then the corresponding matrix series is given by

.
(6.2.1)
If the matrix series is a power series then we will be considering powers of matrices and
hence in this case the series will be defined only for n×n matrices. For an n×n matrix A
consider the series

(6.2.2)
where a0,a1,... are scalars. This is a matrix power series. As in the case of scalar series,
convergence of a matrix series will be defined in terms of the convergence of the sequence
of partial sums.
Definition 6.2.2. Convergence of a matrix series. Let f(A) be a matrix series as in (6.2.1).
Consider the partial sums S0,S1,... where
Sk = A0 + A1 + ... + Ak.
If the sequence S0,S1,... is convergent then we say that the series in (6.2.1) is convergent.
[If it is a power series as in (6.2.2) then Ak = akAk and then the above definition applies].
Example 6.2.3. Check the convergence of the series f1(A) and f2(B) where

43
and

.
Solution 6.2.3. The sum of the first m + 1 terms in f1(A) is given by

.
Convergence of the series in f1(A) depends upon the convergence of the individual
elements in Sm as m → ∞. Note that
∞
Xk 2
y = 1 + y + y + ...
k=0
= (1 − y)−1 if |y| < 1 and + ∞ if y ≥ 1;
= 2;
2
;
Hence the series in f1(A) is convergent for |y| < 1
and diverges if y ≥ 1. Now, consider f2(B). The
partial sums are, for m = 0,1,...,

But = 0 for all m whereas oscillates between 0 and 1 and hence

the sequence of partial sums for this series is not convergent. Thus the series in f2(B) is
not convergent.
Example 6.2.4. Check for the convergence of the following series in the n × n matrix A:
f(A) = I + A + A2 + ...
Solution 6.2.4. Let λ1,λ2,...,λn be the eigenvalues of A. Let us consider the case when the
eigenvalues of A are distinct. Then there exists a nonsingular matrix Q such that
Q−1AQ = D = diag(λ1,...,λn)
and
Q−1AmQ = Dm = diag(

44
Then
Q−1f(A)Q = I + D + D2 + ....
The j-th diagonal element on the right is then
1 + λj + λ2j + ... = (1 − λj)−1 if |λj| < 1, j = 1,...,n
which are the eigenvalues of (I − A)−1. Then if |λj| < 1 for j = 1,2,...,n the series is
convergent and the sum is (I − A)−1 or
I + A + A2 + ... = (I − A)−1 for |λj| < 1, j = 1,...,n.
We can also derive the result from (6.1.5) of Section 6.1. The result also holds good even
if some eigenvalues are repeated. We can state the exponential and trigonometric series
as follows: For any n × n matrix A,

e
(6.2.3)
k=0
and further, when the eigenvalues λ1,...,λn of A are such that |λj| < 1, j = 1,...,n then the
binomial and logarithmic series are given by the following:

.
(6.2.4)
6.2.3. Matrix hypergeometric series
A general hypergeometric series pFq(·) in a real scalar variable x is defined as follows:

(6.2.5)
where, for example,
(a)m = a(a + 1)...(a + m − 1), (a)0 = 1, a 6= 0.
For example,
0F0( ; ;
for |x| < 1.

45
In (6.2.5) there are p upper parameters a1,...,ap and q lower parameters b1,...,bq. The series
in (6.2.5) is convergent for all x if q ≥ p, convergent for |x| < 1 if p = q+1, divergent if p
> q +1 and the convergence conditions for x = 1 and x = −1 can also be worked out. A
matrix series in an n×n matrix A, corresponding to the right side in (6.2.5) is obtained by
replacing x by A. Thus we may define a hypergeometric series in an n × n matrix A as
follows:

(6.2.6)
where a1,...,ap,b1,...,bq are scalars. The series on the right in (6.2.6) is convergent for all A
if q ≥ p, convergent for p = q+1 when the eigenvalues of A are all less than 1 in absolute
value, and divergent when p > q + 1.
Example 6.2.5. If possible, sum up the series

where

.
Solution 6.2.5. Consider the scalar series

= (1 − x)−3 for |x| < 1.

In our matrix A, the eigenvalues are and therefore |λj| <

1, j = 1,2,3. Hence the series can be summed up into a 1F0 type hypergeometric series or
a binomial series and the sum is then
.
But
and

46
6.2.4. The norm of a matrix
For a 1 × 1 vector or a scalar quantity α the absolute value, |α|, is a measure of its
magnitude. For an n × 1 vector X, X0 = (x1,...,xn),

,
(6.2.7)
where |xj| denotes the absolute value of xj, j = 1,...,n, and this can be taken as a measure
of its magnitude. (6.2.7) is its Euclidean length also. This Euclidean length satisfies some
interesting properties.
(a) kXk ≥ 0 for all X
and kXk = 0 if and only if X = O (null);
(b) kαXk = |α| kXk
where α is a scalar quantity;
(c) kX + Y k ≤ kXk + kY
k, the triangular inequality. (6.2.8)
If (a),(b),(c) are taken as postulates or axioms to define a norm of the vector X, denoted
by kXk, then one can see that, not only the Euclidean length but also other items satisfy
(a),(b),(c).
Definition 6.2.3. Norm of a vector and distance between vectors. For X and n×1 vector,
or an element in a general vector subspace S where a norm can be defined, a measure
satisfying (a),(b),(c) above will be called a norm of X and it will be denoted by kXk. Note
that X replaced by X −Y and satisfying (a),(b),(c) is called a distance between X and Y .
It is not difficult to show that the following measures are also norms of the vector X:

( the Euclidean norm )

where X∗ denotes the complex conjugate transpose of X

47
 1 ( the H¨older norms )
kXk∞ = max |xj| ( the infinite norm ).
(6.2.9)
1≤j≤n
Example 6.2.6. Show that kXk1 satisfies the conditions (a),(b),(c) in (6.2.8).
Solution 6.2.6. |xj| being the absolute value of xj cannot be zero unless xj itself is zero. If
xj 6= 0 then |xj| > 0 by definition whether xj is real or complex. Thus condition (a) is
obviously satisfied. Note that for any two scalars α and xj, |αxj| = |α| |xj|. Hence (b) is
satisfied. Also for any two scalars xi and yj the triangular inequality holds. Thus kXk1
satisfies (a),(b),(c) of (6.2.8).
The following properties are immediate from the definition itself .
| kXk − kY k | ≤ kX + Y k ≤ kXk + kY k
k − Xk = kXk
If kXk is a norm of X then kkXk, k > 0 is also a norm of X
| kXk − kY k | ≤ kX − Y k
kUk2 = kXk2 where U = AX, A is a unitary matrix (orthonormal if real) (f) kXk1 ≥ kXk2
≥ ... ≥ kXk∞.
Now let us see how we can define a norm of a matrix as a single number which should
have the desirable properties (a),(b),(c) of (6.2.8). But there is an added difficulty here. If
we consider two matrices, an n × n matrix A and an n × 1 matrix X, then AX is again an
n×1 matrix which is also an n-vector. Hence any definition that we take for the norm of a
matrix must be compatible with matrix multiplication. Therefore an additional postulate
is required.
Definition 6.2.4. A norm of a matrix A. A single number, denoted by kAk, is called a norm
of the matrix A if it satisfies the following four postulates:
kAk ≥ 0 and kAk = 0 if and only if A is a null matrix
kcAk = |c| kAk when c is a scalar
kA + Bk ≤ kAk + kBk whenever A + B is defined (d) kABk ≤ kAk kBk
whenever AB is defined.
It is not difficult to see that the following quantities qualify to be the norms of the matrix
A = (aij):

48
 ,
(6.2.10)
r (H¨older norm, not a norm for p > 2)
(Euclidean norm),
(6.2.11)
(6.2.12)
(6.2.13)
(6.2.14)
(6.2.15)
where s1 is the largest singular value of A;

(6.2.16)
where kAXk and kXk are vector norms, the same norm;

(6.2.17)
same vector norm is taken in each case. As a numerical example let us consider the
following matrix:

.
Then
√ √
kAk1 = |(1 + i)| + |(0)| + |(1)| + |(−1)| =
2+0+1+1=2+ 2;

= 2;
√ √
kAk3 = 2max(
2,0,1,1)] = 2 2;

2;
For computing kAk6 we need the eigenvalues of A∗A.

49
√
The eigenvalues of A∗A are 2 ± 2 and then the largest singular value of A is

.
Note that there are several possible values for kAk7 and kAk8 depending upon which vector
norm is taken. For example, if we take the Euclidean norm and consider kAk8 then it is a

matter of maximizing [ subject to the condition X∗X = 1 where Y = AX. But Y ∗Y

= X∗A∗AX. The problem reduces to the following:
Maximize X∗A∗AX subject to the condition X∗X = 1.
This is already done in Section 5.5 and the answer is the largest eigenvalue of A∗A and
hence, when this particular vector norm is used,
kAk8 = s1 = largest singular value of A.
Note that for a vector norm kXk, kkXk is also a vector norm when k > 0. This property
need not hold for a matrix norm kAk due to condition (d) of the definition.
Example 6.2.7. For an n × n matrix A = (aij) let α = maxi.j |aij|, that is, the largest of the
absolute values of the elements. Is this a norm of A?
Solution 6.2.7. Obviously conditions (a),(b),(c) of Definition 6.2.4 are satisfied. Let us
check condition (d). Let B = (bij) and AB = C = (cij). Then

.
Suppose that the elements are all real and positive and that the largest ones in A and B are
a11 = a and b11 = b. Then
max|aij| = a, max|bij| = b, [max|aij|][max|bij|] = ab i,j i,j i,j i,j
whereas
n max|Xaikbjk| = ab + δ, δ ≥ 0. i,j
k=1
Hence condition (d) is evidently violated. Thus α cannot be a norm of the matrix A. It is
easy to note that β = nα is a norm of A, or
β = nα = n max|aij| = kAk3. (6.2.18)
i,j
Example 6.2.8. Let µA = maxi |λi| where λ1,...,λn be the eigenvalues of an n × n matrix A.
Evidently µ is not a norm of A since condition (a) of Definition 6.2.4 is not satisfied by

50
µ. [Take a non-null triangular matrix with the diagonal elements zeros. Then all
eigenvalues are zeros]. Show that for any matrix norm kAk,
kAk ≥ µA. (6.2.19)
This µA is called the spectral radius of the matrix A.
Solution 6.2.8. Let λ1 be the eigenvalue of A such that µA = λ1. Then, by definition, there
exists a non-null vector X such that
AX1 = λ1X1.
Consider the n × n matrix
B = (X1,O,...,O).
Then
AB = (AX1,O,...,O) = (λ1X1,O,...,O) = λ1B.
From conditions (a) and (d) of Definition 6.2.4
|λ1| kBk ≤ kAk kBk ⇒ kAk ≥ |λ1|
since kBk 6= 0 due to the fact that X1 is non-null. This establishes the result. The result in
(6.2.19) is a very important result which establishes a lower bound for norms of a matrix,
whatever be the norm of a matrix.
6.2.5. Compatible norms
For any n×n matrix A and n×1 vector X if we take any matrix norm kAk and any vector
norm kXk then condition (d) of the definition, namely,
kAXk ≤ kAk kXk
(6.2.20)
need not be satisfied.
Definition 6.2.5. For any matrix A and any vector X, where AX is defined, if (6.2.20) is
satisfied for a particular norm kAk of A and kXk of X then kAk and kXk are called
compatible norms.
It is not difficult to show that the following are compatible norms:
Matrix norm Vector norm
kAk4 of (6.2.13) kXk∞ of (6.2.9)
kAk5 of (6.2.14) kXk1 of (6.2.9)
kAk6 of (6.2.15) kXk2 of (6.2.9)
kAk7 with any vector norm kXkv
kXkv
kAk8 with any vector norm kXk
kXk

51
Example 6.2.9. Show that kAk4 of (6.2.13) and kXk∞ of (6.2.9) are compatible norms.
Solution 6.2.9. Let X be an n × 1 vector with kXk∞ = 1. Consider the vector norm

which establishes the compatibility.

6.2.6. Matrix power series and rate of convergence
Let A be an n × n matrix and consider the power series
f(A) = I + A + A2 + ...
(6.2.21)
We have already seen that the power series in (6.2.21) is convergent when all the
eigenvalues of A are less than 1 in absolute value, that is, 0 < |λj| < 1, j = 1,...,n where the
λj’s are the eigenvalues of A. If kAk denotes a norm of A then evidently
kAkk = kAA...Ak ≤ kAkk.
Then from (6.2.21) we have

Therefore if the power series in (6.2.21) is approximated by taking the first k terms, that
is,
f(A) ≈ I + A + ... + Ak−1
(6.2.22)
then the error in this approximation is given by

(6.2.23)
Thus a measure of an upper bound for the error in the approximation in (6.2.22) is given
by (6.2.23).
6.2.7. An application in statistics
In the field of design of experiments and analysis of variance, connected with two-way
layouts with multiple observations per cell, the analysis of the data becomes quite
complicated when the cell frequencies are unequal. Such a situation can arise, for
example, in a simple randomized block experiment with replicates (the experiment is
repeated a number of times under identical conditions). If some of the observations are
missing in some of the replicates then in the final two-way layout (blocks versus

52
treatments) the cell frequencies will be unequal. In such a situation, in order to estimate
the treatment effects or block effects (main effects) one has to solve a singular system of
a matrix equation of the following type: (This arises from the least square analysis).
(I − A)αˆ = Q
(6.2.24)
where α0 = (α1,...,αp) are the block effects to be estimated, ˆα denotes the estimated value,
A is a p × p matrix

,
and Q is a known column vector. The matrix A is the incidence matrix of this design.
From the design itself αj’s satisfy the condition
α1 + α2 + ... + αp = 0.
(6.2.25)
Observe that A is a singular matrix (the sum of the elements in each row is 1).
Obviously we cannot write and expand
αˆ = (I − A)−1Q = [I + A + A2 + ...]Q
due to the singularity of A. Let k1,...,kp be the medians of the elements in the first, second,
..., p-th rows of A and consider a matrix B = (bij), bij = (aij − ki) for all i and j. Evidently (I
− B) is nonsingular. Consider
(I − B)αˆ = (I − A − K)αˆ = (I − A)αˆ + Kαˆ
where K is a matrix in which all the elements in the i-th row are equal to ki, i = 1,...,p.
Then with (6.2.25) we have Kα = O and hence
(I − A)αˆ = (I − B)αˆ = Q ⇒ αˆ = (I − B)−1Q = (I + B + B2 + ...)Q.
Take the norm kBk4 of (6.2.13). That is,
p
kBk4 = maxX|bij − ki|.
i
j=1
Since the mean deviation is least when the deviations are taken from the median kBk4 is
the least possible for the incidence matrix A so that the convergence of the series I +B +B2
+... is made the fastest possible. In fact, for all practical purposes of testing statistical
hypotheses on αj’s a good approximation is available by taking

53
αˆ ≈ (I + B)Q
where inversion or taking powers of B is not necessary. For an application of the above
procedure to a specific problem in testing of statistical hypothesis see Mathai (1965) [An
approximate method of analysis for a two-way layout, Biometrics, 21,
376-385].

54
CHAPTER 7: SOLUTION OF SYSTEM OF LINEAR EQUATION BY
MATRIX METHOD

7.1 Solution of the linear system AX= B

We now study how to find the solution of system of m linear equations in n unknowns.
Consider the system of equations in unknowns as

………………………………………………….

is called system of linear equations with n unknowns

. If the constants are all zero then
the system is said to be homogeneous type.
The above system can be put in the matrix form as
AX= B

Where X= B=

The matrix is called coefficient matrix, the matrix X is called matrix

of unknowns and B is called as matrix of constants, matrices X and B are of
order .
Definition 7.1: (consistent)
A set of values of which satisfy all these equations
simultaneously is called the solution of the system. If the system has at least
one solution then the equations are said to be consistent otherwise they are
said to be inconsistent. Theorem 5.2:
A system of m equations in n unknowns
represented by the matrix equation AX= B is
)
consistent if and only if . That is the rank of
matrix A is equal to rank of augment matrix (
Theorem 7.2:
If A be an non-singular matrix, X be an matrix and B be an matrix then the
system of equations AX= B has a unique solution.

55
 (1)

Consistent if Inconsistent if

Unique solution if r=n Infinite solutionn if r<n

(2)

If Trivial solution If Infinite solutions

Therefore, every system of linear equations solutions under one of the following:
(i) There is no solution
(ii) There is a unique solution
(iii) There is more than one solution
Methods of solving system of linear Equations:
• Method of intersession:
Consider the matrix equation
Consider the matrix equation
Where
Pre multiplying by , we have

Thus , has only one solution if and is given by .

• Using Elementary row operations: (Gaussian Elimination)
Suppose the coefficient matrix is of the type . That is we have m
equations in n unknowns Write matrix and reduce it to Echelon
augmented form by applying elementary row transformations only.

Example 5.1: Solve the following system of linear equations using matrix method

56
 (ii)

Example 7.2: Determine the values of a so that the following system in unknowns x, y
and z has
(i) No solutions
(ii) More than one solutions
(iii) A unique solution

57
CHAPTER 8: EIGEN VALUES AND EIGENVECTORS:

If A is a square matrix of order n and X is a vector in , ( X considered as column

matrix), we are going to study the properties of non-zero X, where AX are scalar
multiples of one another. Such vectors arise naturally in the study of vibrations, electrical
systems, genetics, chemical reactions, quantum mechanics, economics and geometry.
Definition 8.1:
If A is a square matrix of order n , then a non-zero vector X in is called eigenvector
of A if for some scalar . The scalar is called an eigenvalue of A, and X is said
to be an eigenvector of A corresponding to .
Remark: Eigen values are also called proper values or characteristic values.

Example 8.1: The vector is an eigenvector of A=

Theorem 6.1: If A is a square matrix of order n and is a real number, then is an
eigenvalue of A if and only if .
Proof: If is an eigenvalue of A, the there exist a non-zero X a vector in such that
.

Where I is a identity matrix of order n.

The equation has trivial solution when if and only if . The equation has non-zero
solution if and only if = 0.
Conversely , if = 0 then by the result there will be a non-zero solution for the
equation,

That is, there will a non-zero X in such that , which shows that is an
eigenvalue of A.
Example 6.2: Find the eigen values of the matrixes

(i) A = (ii) B

Theorem 8.2:
If A is an matrix and is a real number, then the following are equivalent:

58
 (i) is an eigenvalue of A.
(ii) The system of equations has non-trivial solutions.
(iii) There is a non-zero vector X in such that .
(iv) Is a solution of the characteristic equation = 0.
Definition 8.2:
Let A bean matrix and be the eigen value of A. The set of all vectors X in
which satisfy the identity is called the eigen space of a corresponding to . This
is denoted by .
Remark:
The eigenvectors of A corresponding to an eigen value are the non-zero vectors of X
that satisfy
. Equivalently the eigen vectors corresponding to are the non zero in the solution
space of
. Therefore, the eigen space is the set of all non-zero X that satisfy
with
trivial solution in addition.
Steps to obtain eigen values and eigen vectors
Step I : For all real numbers form the matrix

Step II: Evaluate That is characteristic polynomial of A.

Step III: Consider the equation ( The characteristic equation of A) Solve

the equation for Let be eigen values of A thus calculated.

Step IV: For each consider the equation

Find the solution space of this system which an eigen space

of A, corresponding to the eigen value of A . Repeat this for each

Step V: From step IV , we can find basis and dimension for each eigen space

for

Example 8.3:

59
 Find (i) Characteristic polynomial
(ii) Eigen values
(iii) Basis for the eigen space of a matrix

Example 8.4:
Find eigen values of the matrix

Also eigen space corresponding to each value of A. Further find basis and dimension for
the same.

.2 Diagonalization:
Definition 6.2.1: A square matrix A is called diagonalizable if there exists an invertible
matrix P such that is a diagonal matrix, the matrix P is said to diagonalizable A.
Theorem 8.2.1: If A is a square matrix of order n, then the following are equivalent.
(i) A is diagonizible.
(ii) A has n linearly independent eigenvectors.
Procedure for diagonalizing a matrix
Step I: Find n linearly independent eigenvectors of A, say
Step II: From the matrix P having as its column vectors.
Step III: The matrix will then be diagonal with as its successive diagonal
.
entries, where is the eigenvalue corresponding to

60
 CHAPTER 9: MATRICES AND LINEAR SYSTEMS
An m×n matrix is a rectangular array of numbers which has m rows and n columns.
We usually put brackets or parentheses around them. Here is a 2 × 3 matrix.

We locate entries in a matrix by specifying its row and column entry. The 1-2 entry (first
row, second column) of the above matrix is 4. In general we index the entries of an
arbitrary n×k matrix like this:

Then the i-j entry of A is denoted by aij, and we can denote the entire matrix by A = (aij).
Note
x1
x2
that a vector ~x ∈ Rn can be interpreted as the n × 1 matrix ... .
xn
Definition: Two matrices are said to be equal if their corresponding entries are equal.
There are two basic binary operations which we will define on matrices – matrix
addition and matrix multiplication.
Definition: Take two matrices A = (aij) and B = (bij). The sum of A and B, is defined to be
the following:

Problem 1 What can you say about the sizes of the matrices A and B as compared to the
size of the matrix A + B? Are there any restrictions?
Problem 2 Prove that matrix addition is both commutative and associative.

This project is adapted from material generously supplied by Prof. Elizabeth Thoren at
UC Santa Barbara.
We now turn our attention to matrix multiplication. The motivation for our definition
comes from our desire to represent a system of linear equations as a matrix multiplication.
Consider the following system of linear equations

61
 x1 + 2x2 + 3x3 = 1
x1 + x2 + x3 = 2
x1 + 4x2 + 7x3 = 1.
From this system we can form the matrix equation A~x = ~b as follows. Note that the
multiplication of a matrix and a vector returns another vector.

We wish to define matrix multiplication such that the above equation is valid (i.e. that the
original system of equations can be recovered from this matrix equation). As another
example, the system
x1 + 2x2 + 3x3 + 4x4 = 1
2x1 + 4x2 + 6x3 + 8x4 = 2
can be expressed as the matrix multiplication

The idea is that a row of the first matrix, times a column of the second matrix gives the
corresponding row-column entry of the product matrix. Notice that the additions become
implicit once we write a system in terms of a matrix.
Here are some more examples.

Definition: Let A be an m × n matrix and B be an n × p matrix. Then the product of A and

B, denoted AB is the m × p matrix whose i-j entry, denoted (AB)ij, is the sum of the
products of corresponding entries from row i of A and column j of B. Thus, we have that
(AB)ij = ai1b1j + ai2b2j + ··· + ainbnj.
Optional (but recommended) Problem: Re-write the definition of matrix multiplication
as a statement involving summation symbols.
Example: Matrix multiplication for 2 × 2 matrices. Let

and
Then

and notice that the columns of and .

62
In fact, given a matrix A and a matrix B whose columns are b~ ~
, if the product
AB is defined then it’s columns are .

Problem 3: Let and . Compute A~x

Problem 4 What can you say about the sizes of the matrices A and B as compared to the
product AB? Are there any restrictions?
Problem 5 Compute the following matrix products:

=?
Problem 6 (a) Find all the solutions to

(b) Find all the solutions to

The matrix in (a) and (b) is said to be row reduced (sometimes called reduced row-echelon
form, or RREF). This concept will be revisited in Theorem 49 of our textbook. Note that
because of its form, it is pretty easy to record the solutions to the systems of equations in
(a) and (b).
Problem 7 Show, by a counterexample, that matrix multiplication is not commutative.
The inverse of a matrix
We have succeeded in transforming the problem “solve a system of linear equations” into
the problem “solve the matrix equation A~x = ~b for the vector ~x.” Treating this matrix
equation algebraically, solving for ~x seems simple. We just need to “divide” each side
by the matrix A. But what does it mean to divide by a matrix? The purpose of this section
is to clarify this.
Definition: The n × n identity matrix, denoted In, is the n × n matrix such that
1 if i = j
( n)ij =
0 if i 6= j
So In has 1’s on the main diagonal, and 0’s everywhere else. As an example

63
Definition: Let A be an n × n matrix. We say that A is invertible if there is a matrix,
denoted A−1, such that AA−1 = In = A−1A. We see that with respect to matrix multiplication
In behaves analogously to how the real number 1 behaves with respect to multiplication
of real numbers. Similarly, the inverse of a matrix behaves much like the reciprocal of a
real number. And just as in the case of real numbers, the inverse of a matrix is unique. In
other words, if a matrix has an inverse, then it has exactly one inverse.
Determinants
Every square matrix, A, has a numerical value associated with it called its determinant,
denoted det(A). We will denote the determinant of a matrix by putting absolute value bars
around the matrix. In this section, we will define the determinant and explore some of its
uses.
Definition: The determinant of a 2 × 2 matrix is the product of the diagonal elements
minus the product of the off-diagonal elements:

Problem 8 Let and .

(a) To show that {~u,~v} spans R2 you must show that any vector can be
represented as a linear combination of ~u and ~v. Write the system of linear
equations that you need to solve, then write this system as a matrix equation.
(b) Solve the system. Does span R2?
(c) Compute det(A) from your matrix equation.

Problem 9 Same problem as Problem 8, but with and .

Problem 10 Now let and . Similarly to the last two problems, to show
that
{~u,~v} spans R2, you must solve the following system for x and y
u1x + v1y = a1
u2x + v2y = a2

Use this idea to prove that = 0 if and only if spans R2.

64
 We usually express the 3 × 3 determinant in terms of the cofactor expansion. The idea
is to compute the determinant by computing the determinants of smaller 2 × 2
submatrices. Notice that

.
Problem 11 Verify that computing the determinant of a 3 × 3 matrix using cofactor
expansion matches the definition of its determinant found in our textbook.
Definition: Let A = (aij) be an n × n matrix. We can define A(i|j) to be the (n − 1) × (n −
1) matrix obtained from A by deleting the ith row and jth column. A(i|j) is called the ijth
maximal submatrix of A.
The cofactor expansions for the determinant give det(A) in terms of the determinants
of the maximal submatrices of A taken along a specific row or column. It turns out that
all of the cofactor expansions give they same value, which is the determinant. So, for
example the determinant can be expressed as the cofactor expansion along the third
column or the cofactor expansion along the second row and both computations will give
the same value:
a12
a22
a32
= −a21det(A(2|1)) + a22det(A(2|2)) − a23det(A(2|3)).
Notice that the sign in front of the term aijdet(Ai|j) is (−1)i+j.
Problem 12 (a) Conjecture a general formula for the cofactor expansion along the 2nd
column of a 3 × 3 matrix.
(b) Test your formula by using it to compute the determinants of the following matrices:

Problem 13 Use cofactor expansion to compute the following

and
In order to make sense of cofactor expansion along a column of a matrix, we need to
explore what happens to the determinant when we flip the entries - i.e. take the transpose
of the matrix:

65
Definition: The transpose of an n × k matrix A, which we denote At, is the k × n matrix
whose columns are formed from the corresponding rows of A. So if A = (aij), then the ijth
entry of At is aji. For example, we have

and .
The second of these two matrices is symmetric.

66
10.0 CONCLUSION AND RECOMMENDATIONS

Conclusion
Matrix algebra has proven to be a fundamental tool in mathematics, engineering, and
computational sciences, with applications spanning diverse fields such as machine
learning, physics, economics, cryptography, and signal processing. The study of
matrices, their transformations, and computational techniques has enabled significant
advancements in data analysis, artificial intelligence, and high-performance computing.
Key findings of this research highlight the importance of eigenvalues, singular value
decomposition (SVD), LU and QR decompositions, and sparse matrix techniques in
optimizing computational efficiency and solving large-scale mathematical problems.
Additionally, technological advancements in parallel computing, GPU acceleration,
and AI-driven optimizations have further enhanced the applicability of matrix
operations in complex problem-solving scenarios.
However, despite the extensive use of matrices in various domains, challenges such as
computational complexity, numerical instability, and scalability continue to persist.
Addressing these issues requires continuous advancements in algorithm development,
error reduction techniques, and efficient computational frameworks to improve the
accuracy and speed of matrix computations.
As matrix algebra remains a cornerstone of modern mathematical and scientific
applications, future research must focus on enhancing computational techniques,
reducing processing time, and integrating emerging technologies like quantum
computing and deep learning.

Recommendations
Based on the findings of this study, the following recommendations are proposed to
enhance the understanding, implementation, and future development of matrix algebra:
1. Optimization of Computational Algorithms – More research should be
conducted on efficient numerical methods to reduce computational costs and
improve the performance of matrix operations, especially for large datasets.
2. Integration with Emerging Technologies – Matrices should be further explored
in the context of quantum computing, cloud computing, and AI-based
optimizations to enhance their application in futuristic computational models.

67
3. Improvement of Numerical Stability – Researchers should develop better error-
handling techniques and conditioning methods to improve numerical stability and
reduce rounding errors in matrix computations.
4. Enhanced Utilization in Machine Learning – As machine learning and deep
learning heavily rely on matrix operations, further exploration of optimized
matrix factorizations and GPU-based accelerations can improve efficiency in
AI applications.
5. Expansion of Matrix Applications in Scientific Fields – Matrix algebra should
be increasingly utilized in biotechnology, climate modeling, and financial risk
assessment, where large-scale data processing is required.
6. Development of Open-Source Computational Libraries – Encouraging the
development of efficient, open-source libraries for matrix computations (such
as Python's NumPy and TensorFlow) will support researchers and developers in
implementing advanced matrix techniques.
7. Education and Training in Matrix Applications – Universities and research
institutions should emphasize the practical applications of matrix algebra in real-
world scenarios to equip students and professionals with necessary computational
skills.
8. Research on Hybrid Matrix Computation Models – Combining classical
numerical methods with AI-driven optimizations and quantum computing
approaches may lead to breakthroughs in solving high-dimensional matrix
problems.

68
11.0 APPENDIX

Appendix A: List of Symbols and Notations

Symbol/Notation Description

A, B, C Matrices

I Identity Matrix

O Zero Matrix

A⁻¹ Inverse of Matrix A

Aᵀ Transpose of Matrix A

det(A) Determinant of Matrix A

tr(A) Trace of Matrix A

λ Eigenvalue

v Eigenvector

UΣVᵀ Singular Value Decomposition (SVD)

LU LU Decomposition

QR QR Decomposition

Appendix B: Glossary of Key Terms

Matrix: A rectangular array of numbers arranged in rows and columns.

Determinant: A scalar value computed from a square matrix.

Eigenvalues and Eigenvectors: Key concepts in linear algebra that describe

transformations.

Singular Value Decomposition (SVD): A method for decomposing a matrix into three
other matrices.

Inverse Matrix: A matrix that results in the identity matrix when multiplied with the
original matrix.

69
LU Decomposition: A technique for factorizing a matrix into lower and upper triangular
matrices.

Sparse Matrix: A matrix in which most of the elements are zero.

QR Decomposition: A method of decomposing a matrix into an orthogonal and an

upper triangular matrix.

Vector Space: A mathematical structure formed by vectors.

Tensor: A generalization of matrices used in deep learning and physics.

Appendix C: Applications of Matrices in Different Fields

Field Application

Machine Learning Neural networks, Principal Component

Analysis (PCA)

Physics Quantum mechanics, Wave equations

Computer Graphics Image processing, Transformations

Economics Markov chains, Portfolio optimization

Engineering Control systems, Circuit analysis

Cryptography Encryption algorithms using matrices

70
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