Matrix (mathematics)
From Wikipedia, the free encyclopedia
Jump to navigationJump to search
For other uses, see Matrix.
"Matrix theory" redirects here. For the physics topic, see Matrix string theory.
An m × n matrix: the m rows are horizontal and the n columns are vertical. Each element of a matrix is
often denoted by a variable with two subscripts. For example, a2,1represents the element at the second row
and first column of the matrix.
In mathematics, a matrix (plural: matrices) is a rectangular array[1] (cf. irregular matrix)
of numbers, symbols, or expressions, arranged in rows and columns.[2][3] For example, the
dimension of the matrix below is 2 × 3 (read "two by three"), because there are two rows and
three columns:
Provided that they have the same size (each matrix has the same number of rows and the
same number of columns as the other), two matrices can be added or subtracted element by
element (see Conformable matrix). The rule for matrix multiplication, however, is that two
matrices can be multiplied only when the number of columns in the first equals the number of
rows in the second (i.e., the inner dimensions are the same, n for an (m×n)-matrix times an
(n×p)-matrix, resulting in an (m×p)-matrix. There is no product the other way round, a first
hint that matrix multiplication is not commutative. Any matrix can be multiplied element-wise
by a scalar from its associated field.
The individual items in an m×n matrix A, often denoted by ai,j, where i and j usually vary
from 1 to m and n, respectively, are called its elements or entries.[4] For conveniently
expressing an element of the results of matrix operations the indices of the element are often
attached to the parenthesized or bracketed matrix expression; e.g.: (AB)i,j refers to an
element of a matrix product. In the context of abstract index notation this ambiguously refers
also to the whole matrix product.
A major application of matrices is to represent linear transformations, that is, generalizations
of linear functions such as f(x) = 4x. For example, the rotation of vectors in three-
dimensional space is a linear transformation, which can be represented by a rotation
matrix R: if v is a column vector (a matrix with only one column) describing the position of a
point in space, the product Rv is a column vector describing the position of that point after a
rotation. The product of two transformation matrices is a matrix that represents
the compositionof two transformations. Another application of matrices is in the solution
of systems of linear equations. If the matrix is square, it is possible to deduce some of its
properties by computing its determinant. For example, a square matrix has an inverse if and
only if its determinant is not zero. Insight into the geometry of a linear transformation is
obtainable (along with other information) from the matrix's eigenvalues and eigenvectors.
Applications of matrices are found in most scientific fields. In every branch of physics,
including classical mechanics, optics, electromagnetism, quantum mechanics, and quantum
electrodynamics, they are used to study physical phenomena, such as the motion of rigid
bodies. In computer graphics, they are used to manipulate 3D models and project them onto
a 2-dimensional screen. In probability theory and statistics, stochastic matrices are used to
describe sets of probabilities; for instance, they are used within the PageRankalgorithm that
ranks the pages in a Google search.[5] Matrix calculus generalizes classical analytical notions
such as derivatives and exponentials to higher dimensions. Matrices are used
in economics to describe systems of economic relationships.
A major branch of numerical analysis is devoted to the development of efficient algorithms
for matrix computations, a subject that is centuries old and is today an expanding area of
research. Matrix decomposition methods simplify computations, both theoretically and
practically. Algorithms that are tailored to particular matrix structures, such as sparse
matricesand near-diagonal matrices, expedite computations in finite element method and
other computations. Infinite matrices occur in planetary theory and in atomic theory. A simple
example of an infinite matrix is the matrix representing the derivative operator, which acts on
the Taylor series of a function.
