EE611
Deterministic Systems
Jordan From, Functions of Square Matrices,
Singular Value Decomposition
Kevin D. Donohue
Electrical and Computer Engineering
University of Kentucky
Eigenvalues and Eigenvectors
A complex (or real) number λ is an eigenvalue of an n
by n matrix A iff ∃ a nonzero vector eigenvector x ∋
A x= x
 A− I  x=0
In order for the nullity of  A− I  to be greater than 0:
∣A− I∣=0
The polynomial resulting from the above determinant
operation is call the characteristic polynomial of A and
is denoted as:
=∣ I−A∣
Examples
Find the eigenvectors and values the following matrices:
[
A= −4 6
−1 1
[
]
−3
0
1
A= 0.25 −2.75 −0.25
0.25 0.25 −2.25
[
A= 4 13
−2 −6
] [
]
−2.75 0.25 0.75
A= 0
−3
0
0.25 0.25 −2.25
]
Jordan-Form Representations
A Jordan form matrix consist of a diagonal matrix (in the case of
distinct eigenvalues) or block diagonal and triangular forms (in the
case of repeated eigenvalues.)
[
[
] [ ]
] [ ]
1 0 0 0
0 2 0 0
A=
0 0 3 0
0 0 0 4
1 0 0 0
0 2 0 0
A=
0 0 2 1
0 0 0 2
1 0 0 0
0 2 1 0
A=
0 0 2 1
0 0 0 2
1 1 0 0
0 1 0 0
A=
0 0 2 1
0 0 0 2
Distinct Eigenvalue Example
If all eigenvalues are distinct then eigenvectors associated with
each eigenvalue are l.i. and can be used form the Q matrix for a
similarity transformation to make the matrix diagonal.
Find the P and Q matrices to diagonalize the matrix below:
[
A= −4 6
−1 1
]
Repeated Eigenvalues
If eigenvalues for an n by n matrix are repeated, then the existence
of n l.i. eigenvectors is not guaranteed. The generalized
eigenvectors can be used in cases where n l.i. eigenvectors cannot
be found.
A generalized eigenvector of grade n for eigenvalue λ satisfies the
following relationships
n
 A− I  v=0
 A− I 
n−1
v≠0
Generalized eigenvectors lead to the block-diagonal and triangular
Jordan forms.
Repeated Eigenvalue Examples
Find the P and Q matrices to put the matrices below in Jordan
form:
[
]
[
]
−3
0
1
A= 0.25 −2.75 −0.25
0.25 0.25 −2.25
−2.75 0.25 0.75
A= 0
−3
0
0.25 0.25 −2.25
Companion-Form Matrices
Verify that companion-form matrices have a characteristic equation
of
=∣ I−A∣=4 1 32 2 3 4
[
0
1
A=
0
0
0
0
1
0
0
0
0
1
− 4
−3
− 2
−1
] [
0
1
0
0
0
1
0
A= 0
0
0
0
1
−4 −3 −2 −1
]
Square Matrix Polynomials
If an N by N matrix is diagonal, raising it to a power is simply
raising the diagonal elements to the same power and can be
decompose into N polynomial equations. If matrix is block
diagonal, then powers can be applied to each block and
decomposed along the blocks.
Apply a similarity transformation to a matrix polynomial and show
that
 Q−1
f A=Q f  A
 Q−1
given
A=Q A
Minimal Polynomials
Given the characteristic polynomial of NM by N matrix:
n
=∣ I−A∣= N  1  N −1 ... N =∏ − i 
i=1
where M is the number of diagonal blocks corresponding to an ni
multiplicity of eigenvalues.
i
Define the index of λi, denoted as ni , as the largest order of all
Jordan blocks associated with λi. Then the minimal polynomial is

M
given by:
n
=∏ −i  
i
i=1
It can be shown that ψ (λ) is the smallest order polynomial such
that
A=0
Cayley-Hamilton Theorem
Given the characteristic polynomial of N by N matrix A:
=∣ I−A∣=N 1  N −1 ... N
Then
A=A N 1 A N−1... N I=0
By noting that minimal polynomial ψ (λ) can be factor out of
characteristics polynomial, the above theorem is established.
This implies that −A N =1 A N−1 2 A N−2 ... N I
and in general any polynomial of A can be expressed in terms of an
N linear combination of {I, A, ... , An-1}
f A= N −1 A N−1 ...1 A0 I
Functions in Terms of Polynomials
Given f(λ) and an N by N matrix A with characteristic polynomial:
M
ni
=∏ −i 
i=1
where M is the number of diagonal blocks corresponding to an ni
multiplicity of eigenvalues. Define N-1 order polynomial:
N −1
h= N −1  ...1 0
f(Α) can be evaluated with h(Α) if the β coefficients are resolved
with the following set of equations:
f l  i =h l i  for l=0,1,...ni −1 and i=1,2,... , M
Functions in Terms of Polynomials
Given
find
[
2
A= 0
−1 −3
]
A10
[
−3
0
1
Given A= 0.25 −2.75 −0.25
0.25 0.25 −2.25
Find exp−2 A
]
Singular Values
Given m by n real matrix H and M = H'H. Since M is symmetric
and semi-positive definite, its eigenvalues are real and positive.
Let r be the number of nonzero eigenvalues of M and rank them:
2
1
2
2
2
r
 ≥ ≥.....≥ 0=...=
the λ values are called the singular values of H
Given
[
A= −1 0 1
2 −1 0
]
Find the singular values of A and A'
2
n
Singular Value Decomposition
Every m by n matrix H can be decomposed into:
H=R S Q '
with R'R = RR' = Im , Q'Q = QQ' = In , and m by n diagonal
matrix S with singular values of H on the diagonal.
Given
[
−3 0
1
A= 1 −1 0
10 10 −2
]
Use the SVD to take the inverse of A