Eigenvalues and Eigenvectors
Eigenvalues and Eigenvectors
Just like the space Rn , we also define the space Cn .
Eigenvalues and Eigenvectors
Just like the space Rn , we also define the space Cn .
Indeed,
Cn = {[x1 , x2 , . . . , xn ]t : x1 , x2 , . . . , xn ∈ C}.
Eigenvalues and Eigenvectors
Just like the space Rn , we also define the space Cn .
Indeed,
Cn = {[x1 , x2 , . . . , xn ]t : x1 , x2 , . . . , xn ∈ C}.
The definitions of vector addition and scalar multiplication
etc., and most of the results that we have studied so far in
case of Rn , can also be accomplished for the space Cn , in
a similar manner.
Definition
Let A be an n × n matrix.
A complex number λ is called an eigenvalue of A if there is
a non-zero vector x ∈ Cn such that Ax = λx.
Definition
Let A be an n × n matrix.
A complex number λ is called an eigenvalue of A if there is
a non-zero vector x ∈ Cn such that Ax = λx.
Such a vector x is called an eigenvector of A
corresponding to λ.
Definition
Let A be an n × n matrix.
A complex number λ is called an eigenvalue of A if there is
a non-zero vector x ∈ Cn such that Ax = λx.
Such a vector x is called an eigenvector of A
corresponding to λ.
Example
1 3
The number 4 is an eigenvalue of A = with
3 1
corresponding eigenvector [1, 1]t .
Definition
Let λ be an eigenvalue of a matrix A. The collection of all
eigenvectors of A corresponding to λ, together with the zero
vector, is called the eigenspace of λ, and is denoted by Eλ .
Definition
Let λ be an eigenvalue of a matrix A. The collection of all
eigenvectors of A corresponding to λ, together with the zero
vector, is called the eigenspace of λ, and is denoted by Eλ .
Result
Let A be an n × n matrix and let λ be an eigenvalue of A. Then
Eλ = null(A − λI), that is, Eλ is a subspace of Cn .
Definition
Let λ be an eigenvalue of a matrix A. The collection of all
eigenvectors of A corresponding to λ, together with the zero
vector, is called the eigenspace of λ, and is denoted by Eλ .
Result
Let A be an n × n matrix and let λ be an eigenvalue of A. Then
Eλ = null(A − λI), that is, Eλ is a subspace of Cn .
λ is an eigenvalue of A iff det(A − λI) = 0.
Definition
Let λ be an eigenvalue of a matrix A. The collection of all
eigenvectors of A corresponding to λ, together with the zero
vector, is called the eigenspace of λ, and is denoted by Eλ .
Result
Let A be an n × n matrix and let λ be an eigenvalue of A. Then
Eλ = null(A − λI), that is, Eλ is a subspace of Cn .
λ is an eigenvalue of A iff det(A − λI) = 0.
Definition
Let A be an n × n matrix. Then
det(A − λI) is called characteristic polynomial of A.
Definition
Let λ be an eigenvalue of a matrix A. The collection of all
eigenvectors of A corresponding to λ, together with the zero
vector, is called the eigenspace of λ, and is denoted by Eλ .
Result
Let A be an n × n matrix and let λ be an eigenvalue of A. Then
Eλ = null(A − λI), that is, Eλ is a subspace of Cn .
λ is an eigenvalue of A iff det(A − λI) = 0.
Definition
Let A be an n × n matrix. Then
det(A − λI) is called characteristic polynomial of A.
det(A − λI) = 0 is called characteristic equation of A.
Method of Finding Eigenvalues and Bases for
Corresponding Eigenspaces
Let A be an n × n matrix.
1 Compute the characteristic polynomial det(A − λI).
Method of Finding Eigenvalues and Bases for
Corresponding Eigenspaces
Let A be an n × n matrix.
1 Compute the characteristic polynomial det(A − λI).
2 Find the eigenvalues of A by solving the characteristic
equation det(A − λI) = 0 for λ.
Method of Finding Eigenvalues and Bases for
Corresponding Eigenspaces
Let A be an n × n matrix.
1 Compute the characteristic polynomial det(A − λI).
2 Find the eigenvalues of A by solving the characteristic
equation det(A − λI) = 0 for λ.
3 For each eigenvalue λ, find null(A − λI) = Eλ .
Method of Finding Eigenvalues and Bases for
Corresponding Eigenspaces
Let A be an n × n matrix.
1 Compute the characteristic polynomial det(A − λI).
2 Find the eigenvalues of A by solving the characteristic
equation det(A − λI) = 0 for λ.
3 For each eigenvalue λ, find null(A − λI) = Eλ .
4 Find a basis for each eigenspace.
Example
Find the eigenvalues and the corresponding eigenspaces of the
following matrices:
0 1 0 −1 0 1
0 0 1 and 3 0 −3 .
2 −5 4 1 0 −1
Example
Find the eigenvalues and the corresponding eigenspaces of the
following matrices:
0 1 0 −1 0 1
0 0 1 and 3 0 −3 .
2 −5 4 1 0 −1
Definition
Let λ be an eigenvalue of a matrix A.
The algebraic multiplicity of λ is the multiplicity of λ as a
root of the characteristic polynomial of A.
Example
Find the eigenvalues and the corresponding eigenspaces of the
following matrices:
0 1 0 −1 0 1
0 0 1 and 3 0 −3 .
2 −5 4 1 0 −1
Definition
Let λ be an eigenvalue of a matrix A.
The algebraic multiplicity of λ is the multiplicity of λ as a
root of the characteristic polynomial of A.
The geometric multiplicity of λ is the dimension of Eλ .
