MAA704:
Matrix
functions and
matrix
equations
MAA704: Matrix functions and matrix
equations
Karl Lundengård
Matrix
functions
Power function
Root function
Matrix
polynomials
Exponential
matrix
Solving matrix
differential
equations
General matrix
functions
Matrix
equations
Tensor products
December 3, 2012
Matrix
equations,
cont’d
Summary
Contents of todays lecture
MAA704:
Matrix
functions and
matrix
equations
Matrix
functions
I
I
Matrix functions
Matrix equations
I
Tensor products (Kroenecker product)
Power function
Root function
Matrix
polynomials
Exponential
matrix
Solving matrix
differential
equations
General matrix
functions
Matrix
equations
Tensor products
Matrix
equations,
cont’d
Summary
Some useful functions from calculus
Power function: f (x) = x n , x ∈ C, n ∈ Z+
√
Root function: f (x) = n x, x ∈ R, n ∈ Z+
n
X
Polynomials: p(x) =
ak x k , a,x ∈ C, n ∈ Z+
k=0
Exponential function: f (x) = e x , x ∈ C
MAA704:
Matrix
functions and
matrix
equations
Matrix
functions
Power function
Root function
Matrix
polynomials
Exponential
matrix
Solving matrix
differential
equations
General matrix
functions
Matrix
equations
Tensor products
Can we make matrix versions of these functions?
Matrix
equations,
cont’d
Summary
MAA704:
Matrix
functions and
matrix
equations
Power function
Matrix
functions
n
I
What could A mean?
I
Natural interpretation:
An = AAA
. . . A}, n ∈ Z+
| {z
(1)
A =I
(2)
n
0
−n
A
−1 n
= (A
+
) , n ∈ Z , A is invertible
A ∈ Mk×k
(3)
(4)
Power function
Root function
Matrix
polynomials
Exponential
matrix
Solving matrix
differential
equations
General matrix
functions
Matrix
equations
Tensor products
Matrix
equations,
cont’d
Summary
Calculating An
MAA704:
Matrix
functions and
matrix
equations
Matrix
functions
I
Direct calculation of An might mean a lot of work,
especially for large n.
I
Is it easier to calculate for certain types of matrices?
I
How can the calculation of An be simplified?
Power function
Root function
Matrix
polynomials
Exponential
matrix
Solving matrix
differential
equations
General matrix
functions
Matrix
equations
Tensor products
Matrix
equations,
cont’d
Summary
Calculating An
MAA704:
Matrix
functions and
matrix
equations
Matrix
functions
I
Diagonal m × m matrix
 n
d11 0 . . .
 0 dn . . .
22

Dn =  .
..
..
.
 .
.
.
0
0 ...
0
0
..
.
n
dmm





Power function
Root function
Matrix
polynomials
Exponential
matrix
Solving matrix
differential
equations
General matrix
functions
Matrix
equations
Tensor products
Matrix
equations,
cont’d
Summary
Calculating An
MAA704:
Matrix
functions and
matrix
equations
Matrix
functions
I
I
Block-diagonal matrix
 n
D1 0 . . .
 0 Dn . . .
2

Dn =  .
..
..
.
 .
.
.
0
0 ...
0
0
..
.
Dkn





Di , 1 ≤ i ≤ k, are square matrices of (different) size.
Power function
Root function
Matrix
polynomials
Exponential
matrix
Solving matrix
differential
equations
General matrix
functions
Matrix
equations
Tensor products
Matrix
equations,
cont’d
Summary
Calculating An
MAA704:
Matrix
functions and
matrix
equations
Matrix
functions
I
D n for diagonal matrices is easy!
I
What about diagonalizable matrices?
I
D n for block diagonal matrices is relatively easy!
I
Any matrix can be written as a Jordan matrix, which is a
block diagonal matrix. Can this be useful?
Power function
Root function
Matrix
polynomials
Exponential
matrix
Solving matrix
differential
equations
General matrix
functions
Matrix
equations
Tensor products
Matrix
equations,
cont’d
Summary
Calculating An
I
Let A and B be similar matrices
A = S −1 BS
I
For positive integer n
An = |AAA{z
. . . A} =
n
=S
−1
B |SS{z−1} B |SS{z−1} BS . . . BS =
=I
=S
−1
=I
n
B S
MAA704:
Matrix
functions and
matrix
equations
Matrix
functions
Power function
Root function
Matrix
polynomials
Exponential
matrix
Solving matrix
differential
equations
General matrix
functions
Matrix
equations
Tensor products
Matrix
equations,
cont’d
Summary
I
I
I
Can be shown in the same way that An = S −1 B n S for any
integer n.
Diagonalizable matrix: An = S −1 D n S
Any square matrix: An = S −1 J n S
Calculating An
MAA704:
Matrix
functions and
matrix
equations
Matrix
functions
I
I
−1
Can be easier to find B than S in A = S BS.
Finding B might be enough since similar matrices share
several properties:
I
I
I
I
Eigenvalues (but generally not eigenvectors)
Determinant
Trace
Rank
Power function
Root function
Matrix
polynomials
Exponential
matrix
Solving matrix
differential
equations
General matrix
functions
Matrix
equations
Tensor products
Matrix
equations,
cont’d
Summary
MAA704:
Matrix
functions and
matrix
equations
Root function
I
What could A =
√
n
1
B = B n mean?
Definition
If the following relation holds for two square matrices
A, B ∈ Mk×k
A = |BBB{z
. . . B}
n
then √
B is said to be the nth root of A. This is annotated
1
n
B = A = An
√
n
I How do you find
B?
√
n
I For how many different A is A =
B
Matrix
functions
Power function
Root function
Matrix
polynomials
Exponential
matrix
Solving matrix
differential
equations
General matrix
functions
Matrix
equations
Tensor products
Matrix
equations,
cont’d
Summary
MAA704:
Matrix
functions and
matrix
equations
Square root
Matrix
functions
√
A=
B ⇔ AA = B
√
I
How do you find
B?
I
For how many different A is A =
√
B?
Power function
Root function
Matrix
polynomials
Exponential
matrix
Solving matrix
differential
equations
General matrix
functions
Matrix
equations
Tensor products
Matrix
equations,
cont’d
Summary
Square root of a diagonal matrix
Theorem
If we have two diagonal square matrices, C and D, such that
√

d11 √0
...
0
 0
d22 . . .
0 


C = .
..
..  ⇒
..
 ..

.
.
√.
dmm
0
0
...


d11 0 . . .
0
 0 d22 . . .
0 


⇒ CC = C 2 =  .
.. . .
.. 
 ..
.
.
. 
0
Then C is a root of D.
0
. . . dmm
MAA704:
Matrix
functions and
matrix
equations
Matrix
functions
Power function
Root function
Matrix
polynomials
Exponential
matrix
Solving matrix
differential
equations
General matrix
functions
Matrix
equations
Tensor products
Matrix
equations,
cont’d
Summary
Square root of a diagonal matrix
I
p
There are at least 2n roots since we can choose ± dii
p
If all dii are chosen positive, this is called the principal
root.
I
Can be many more roots
I
Theorem
1 0
All of the following matrices are roots to I2 =
0 1
1 ∓s ∓r
1 ±s ∓r
±1 0
1 0
,
,
,
0 1
0 ±1
t ∓r ±s
t ∓r ∓s
if r 2 + s 2 = t 2 .
MAA704:
Matrix
functions and
matrix
equations
Matrix
functions
Power function
Root function
Matrix
polynomials
Exponential
matrix
Solving matrix
differential
equations
General matrix
functions
Matrix
equations
Tensor products
Matrix
equations,
cont’d
Summary
Square root of similar matrices
MAA704:
Matrix
functions and
matrix
equations
Matrix
functions
Let A = S −1 BS then
√
√
A = S −1 BS
Power function
Root function
Matrix
polynomials
Exponential
matrix
Solving matrix
differential
equations
General matrix
functions
Matrix
equations
Tensor products
Matrix
equations,
cont’d
Summary
MAA704:
Matrix
functions and
matrix
equations
Square root of a Jordan matrix
Matrix
functions
p
Jm1 (λ1 ) p 0

0
Jm2 (λ2 )
√

.
..
J=
..

.

0
0
...
...
..
.
0
0





.
.
.

q
Jmk (λk )
...
Power function
Root function
Matrix
polynomials
Exponential
matrix
Solving matrix
differential
equations
General matrix
functions
Matrix
equations
Tensor products
Matrix
equations,
cont’d
Summary
Square root of a Jordan block
MAA704:
Matrix
functions and
matrix
equations
Matrix
functions

λ 1 0 ...
0 λ 1 . . .


Jm (λ) =  0 0 λ . . .
 .. .. .. . .
. . .
.
0 0 0 ...
Power function
Root function
Matrix
polynomials
Exponential
matrix
Solving matrix
differential
equations
General matrix
functions
 √


1
0
λ √0 . . . 0
0 1
0
λ . . . 0




0
1 . . . 0
 = λ(I +K )
 = λ 0 0
.

.. 
.
.
.
.
Matrix
.
.
.
.
.
.
.
. .
.
.
equations
Tensor products
λ
0 0
0 ... 1
Matrix
equations,
cont’d
Summary
Square root of a Jordan block
I
K is a strictly triangular matrix
I
K is nilpotent ⇔ K n = 0 for some finite integer n ≥ 0.
Theorem
p
√
1
1
1
5 4
Jm (λ) = I + K = 1 + K − K 2 + K 3 −
K + ...
2
8
16
128
Proof.
See compendium page 61.
MAA704:
Matrix
functions and
matrix
equations
Matrix
functions
Power function
Root function
Matrix
polynomials
Exponential
matrix
Solving matrix
differential
equations
General matrix
functions
Matrix
equations
Tensor products
Matrix
equations,
cont’d
Summary
Matrix polynomial
MAA704:
Matrix
functions and
matrix
equations
Matrix
functions
Definition
A matrix polynomial of degree n is a matrix function with this
form
n
X
p(A) =
ck Ak , c ∈ K, A ∈ Mn×n (K)
k=0
Power function
Root function
Matrix
polynomials
Exponential
matrix
Solving matrix
differential
equations
General matrix
functions
Matrix
equations
Tensor products
Matrix
equations,
cont’d
Summary
MAA704:
Matrix
functions and
matrix
equations
Interesting properties of the matrix polynomial
I
I
I
I
I
Matrix
functions
The coefficients behave like the coefficients for regular
polynomials
p(x) + q(x) has the same coefficients as p(A) + q(A)
p(x)q(x) has the same coefficients as p(A)q(A)
p(S −1 AS) = S −1 p(A)S

 
A11 0 . . . 0
p(A11 )
0
 0 A22 . . . 0   0
p(A22 )

 
p  .
.. . . ..  =  ..
..
.
 .
. .   .
.
.
0 0 . . . Akk
0
0
where all Aii are quadratic blocks.
...
...
..
.
0
0
..
.





. . . p(Akk )
Power function
Root function
Matrix
polynomials
Exponential
matrix
Solving matrix
differential
equations
General matrix
functions
Matrix
equations
Tensor products
Matrix
equations,
cont’d
Summary
The Cayley-Hamilton theorem
MAA704:
Matrix
functions and
matrix
equations
Matrix
functions
Theorem (Cayley-Hamilton theorem)
Let pA (λ) = det(λI − A) then pA (A) = 0
Power function
Root function
Matrix
polynomials
Exponential
matrix
Solving matrix
differential
equations
General matrix
functions
Matrix
equations
Tensor products
Matrix
equations,
cont’d
Summary
MAA704:
Matrix
functions and
matrix
equations
The adjugate formula
Theorem (The adjugate formula)
Matrix
functions
For any square matrix A the following equality holds
A adj(A) = adj(A)A = det(A)I
where adj(A) is a square matrix such that
adj(A)ki = Aki
(5)
Power function
Root function
Matrix
polynomials
Exponential
matrix
Solving matrix
differential
equations
General matrix
functions
Matrix
equations
Tensor products
where Aki is the ki-cofactor of A
Matrix
equations,
cont’d
Proof.
Summary
See compendium page 62-64.
Sketch of proof for Cayley-Hamilton
Proof.
Matrix
functions
pA (λ)I = (λI − A)B
B=
n−1
X
Power function
Root function
Matrix
polynomials
Exponential
matrix
Solving matrix
differential
equations
General matrix
functions
λk Bk
k=0
(λI − A)B = (λI − A)
n−1
X
MAA704:
Matrix
functions and
matrix
equations
k
λ Bk =
k=0
Matrix
equations
Tensor products
Matrix
equations,
cont’d
= λn Bn−1 + λn−1 (Bn−2 − ABn−1 ) + . . . + λ(B0 − AB1 ) + (−AB0 )Summary
=
= λn I + λn−1 an−1 I + . . . + a1 λ + a0 = pA (λ)I
MAA704:
Matrix
functions and
matrix
equations
Sketch of proof for Cayley-Hamilton
Proof.
If this expression holds for all λ then the following three
conditions must be fulfilled
ak I = Bk−1 − ABk for 1 ≤ k ≤ n − 1
Matrix
functions
(6)
I = Bn−1
(7)
a0 I = −AB0
(8)
Next consider the polynomial pA (A)
Power function
Root function
Matrix
polynomials
Exponential
matrix
Solving matrix
differential
equations
General matrix
functions
Matrix
equations
Tensor products
pA (A) = An + An−1 an−1 + . . . + Aa1 + Ia0
Matrix
equations,
cont’d
Combining this with condition (6)-(8) results in
Summary
pA (A) = An + An−1 (Bn−2 − ABn−1 ) + . . . + A(B0 − AB1 ) + (−AB0 ) =
= An (I − Bn−1 ) +An−1 (Bn−2 − Bn−2 ) + . . . + A (B0 − B0 ) = 0
| {z }
|
{z
}
| {z }
=0
=0
=0
MAA704:
Matrix
functions and
matrix
equations
Companion matrix
Matrix
functions
Definition
n
n−1
Let p(x) = x + an−1 x
+ . . . + a1 x + a0 then


0
1
0
...
0
0
 0
0
1
...
0
0 


 ..
..
..
..
.. 
..
C (p) =  .
.
.
.
.
. 


 0
0
0
...
0
1 
−a0 −a1 −a2 . . . −an−2 −an−1
Power function
Root function
Matrix
polynomials
Exponential
matrix
Solving matrix
differential
equations
General matrix
functions
Matrix
equations
Tensor products
Matrix
equations,
cont’d
Summary
MAA704:
Matrix
functions and
matrix
equations
State variable representation
I
Consider the linear differential equation (with constant
scalar coefficients) with y = y (t)
y
I
(n)
+ an−1 y
(n−1)
0
+ . . . + a1 y + a0 = f
Create the state variable vector


y
 y0 


 . 
x =  .. 
 n−2 
y

n−1
y
Matrix
functions
Power function
Root function
Matrix
polynomials
Exponential
matrix
Solving matrix
differential
equations
General matrix
functions
Matrix
equations
Tensor products
Matrix
equations,
cont’d
Summary
State variable representation
I
Rewrite differential equation as a linear equation system
 0
= x2

 x10


x
= x3

 2
..
.


0

x
=
xn


 n−1
= −a0 x1 − a1 x2 − . . . − an−1 xn + f
xn0
MAA704:
Matrix
functions and
matrix
equations
Matrix
functions
Power function
Root function
Matrix
polynomials
Exponential
matrix
Solving matrix
differential
equations
General matrix
functions
Matrix
equations
Tensor products
I
Rewrite on matrix form
dx
= C (p)x
dt
with p(x) = x n + an−1 x n−1 + . . . + a1 x + a0
Matrix
equations,
cont’d
Summary
State variable representation
MAA704:
Matrix
functions and
matrix
equations
Matrix
functions
I
Note that det(λI − C (p)) = p(λ) which means that for
any λ ∈ Sp(C (p)) p(λ) = 0.
I
Can be a convenient way of finding roots to a polynomial
which can then be used to solve a differential equation.
Power function
Root function
Matrix
polynomials
Exponential
matrix
Solving matrix
differential
equations
General matrix
functions
Matrix
equations
Tensor products
Matrix
equations,
cont’d
Summary
First order differential matrix equations
MAA704:
Matrix
functions and
matrix
equations
Matrix
functions
dU
= AU, U(0) = I , U, A ∈ Mn×n
dt
I
How do you solve this equation?
I
If A had not been a matrix then the answer would be
U = e At , what is the corresponding matrix answer?
Power function
Root function
Matrix
polynomials
Exponential
matrix
Solving matrix
differential
equations
General matrix
functions
Matrix
equations
Tensor products
Matrix
equations,
cont’d
Summary
Exponential matrix
I
Instead of shoving matrices into the definition and hoping
they fit we can construct matrix equations that have some
desirable property of a normal function of numbers.
I
The exponential function e at has three nice properties
d at
(e ) = ae at = e at a
dt
(e at )−1 = e −at
e a · e b = e a+b = e b+a = e b · e a
MAA704:
Matrix
functions and
matrix
equations
Matrix
functions
Power function
Root function
Matrix
polynomials
Exponential
matrix
Solving matrix
differential
equations
General matrix
functions
Matrix
equations
Tensor products
Matrix
equations,
cont’d
Summary
Exponential matrix
MAA704:
Matrix
functions and
matrix
equations
Matrix
functions
Theorem
The following is true for the exponential matrix
d tA
e = Ae tA = e tA A
a)
dt
b) (e tA )−1 = e −tA
c) e A+B = e A e B = e B e A if AB = BA
Power function
Root function
Matrix
polynomials
Exponential
matrix
Solving matrix
differential
equations
General matrix
functions
Matrix
equations
Tensor products
Matrix
equations,
cont’d
Summary
Exponential matrix
Consider what happens to a general polynomial of At when we
take the first time derivative
p(At) =
n
X
k=0
ak (At)k ⇔
d
(p(At)) =
dt
n
X
ak kAk t k−1 =
k=1
n−1
X
=A
al+1 (l + 1)Al t l
l=0
MAA704:
Matrix
functions and
matrix
equations
Matrix
functions
Power function
Root function
Matrix
polynomials
Exponential
matrix
Solving matrix
differential
equations
General matrix
functions
Matrix
equations
Tensor products
The last equality is achieved by simply taking l = k − 1.
Compare this to
Matrix
equations,
cont’d
Summary
Ap(At) = A
n
X
l=0
al Al t l
MAA704:
Matrix
functions and
matrix
equations
Exponential matrix
Matrix
functions
If the polynomial coefficients are chosen such that
al+1 (l + 1) = al these expressions would be identical expect for
the final term. Let us choose the correct coefficients
al+1 (l + 1) = al ⇔ al =
1
l!
and simply avoid the problem of the last term not matching up
by making the polynomial have infinite degree.
Power function
Root function
Matrix
polynomials
Exponential
matrix
Solving matrix
differential
equations
General matrix
functions
Matrix
equations
Tensor products
Matrix
equations,
cont’d
Summary
MAA704:
Matrix
functions and
matrix
equations
Exponential matrix
Matrix
functions
Definition
The matrix polynomial function is defined as
e
At
=
∞ k
X
t
k=0
k!
Ak , A ∈ Mn×n
Power function
Root function
Matrix
polynomials
Exponential
matrix
Solving matrix
differential
equations
General matrix
functions
Matrix
equations
Tensor products
Matrix
equations,
cont’d
Summary
Calculating the exponential matrix
I
Calculating the exponential matrix e A is generally difficult.
I
Some classes of matrices are simple, for example diagonal
matrices
 d11

e
0 ... 0
 0 e d22 . . . 0 


D
e = .
..
.. 
.
.
.
 .
.
.
. 
0
0
. . . e dnn
MAA704:
Matrix
functions and
matrix
equations
Matrix
functions
Power function
Root function
Matrix
polynomials
Exponential
matrix
Solving matrix
differential
equations
General matrix
functions
Matrix
equations
I
This means the exponential of a diagonalizable matrix can
be calculated like this
e A = S −1 e D S
Prove this as an exercise. Hint: What is the Taylor series
(power series) for e at ?
Tensor products
Matrix
equations,
cont’d
Summary
1st order differential matrix equation
Theorem
The exponential matrix e xA is the solution to the initial value
problem

 dU = AU
dx

U(0) = I
MAA704:
Matrix
functions and
matrix
equations
Matrix
functions
Power function
Root function
Matrix
polynomials
Exponential
matrix
Solving matrix
differential
equations
General matrix
functions
Matrix
equations
Proof.
Tensor products
xA
From the construction of e we already know that if fulfills
the differential equation. Checking that the initial condition is
fulfilled can be done simply by setting x = 0 in the
definition.
Matrix
equations,
cont’d
Summary
MAA704:
Matrix
functions and
matrix
equations
Matrix
functions
I
Can we systematically create a matrix function for every
function of numbers?
Power function
Root function
Matrix
polynomials
Exponential
matrix
Solving matrix
differential
equations
General matrix
functions
Matrix
equations
Tensor products
Matrix
equations,
cont’d
Summary
MAA704:
Matrix
functions and
matrix
equations
General matrix functions
Definition (Matrix function via Hermite interpolation)
If a function f (λ) is defined on all λ that are eigenvalues of a
square matrix A then the matrix function f (A) is defined as
f (A) =
h sX
i −1 (k)
X
f (λi )
i=1 k=0
k!
φik (A)
Matrix
functions
Power function
Root function
Matrix
polynomials
Exponential
matrix
Solving matrix
differential
equations
General matrix
functions
Matrix
equations
Tensor products
where φik are the Hermite interpolation polynomials (defined
below), λi are the different eigenvalues and si is the multiplicity
of each eigenvalue.
Matrix
equations,
cont’d
Summary
General matrix functions
MAA704:
Matrix
functions and
matrix
equations
Matrix
functions
Definition (Hermite interpolation polynomials)
Let lambdai , 1 ≤ i ≤ n be complex numbers and si , 1 ≤ i ≤ n,
be positive integers. Let s be the sum of all si . The Hermitian
interpolation polynomials are a set of polynomials with degree
lower than s that fulfills the conditions
(
(l)
0 if i 6= j or k 6= l
φik (λj )
=
l!
1 if i = j and k = l
Power function
Root function
Matrix
polynomials
Exponential
matrix
Solving matrix
differential
equations
General matrix
functions
Matrix
equations
Tensor products
Matrix
equations,
cont’d
Summary
MAA704:
Matrix
functions and
matrix
equations
General matrix functions
Definition (Matrix function via Cauchy Integral)
Let C be a curve that encloses all eigenvalues of the square
matrix A. If the function f is analytical (can be written as a
power series) on C and inside C then
Z
1
f (z)(zI − A)−1 dz
f (A) =
2πi C
−1
The matrix RA (z) = (zI − A)
of A.
I
is called the resolvent matrix
Equivalent to the Hermite interpolation definition (for
analytical functions).
Matrix
functions
Power function
Root function
Matrix
polynomials
Exponential
matrix
Solving matrix
differential
equations
General matrix
functions
Matrix
equations
Tensor products
Matrix
equations,
cont’d
Summary
MAA704:
Matrix
functions and
matrix
equations
Matrix power series
I
Matrix
functions
Using Taylor expansion
f (x) =
∞
X
f (n) (a)
n=0
n
(x − a)n
many functions (to be precise, all analytical functions) can
be rewritten as a (convergent) polynomial of infinite
degree.
I
This can be proved from the Cauchy integral of the
resolvent.
Power function
Root function
Matrix
polynomials
Exponential
matrix
Solving matrix
differential
equations
General matrix
functions
Matrix
equations
Tensor products
Matrix
equations,
cont’d
Summary
Matrix equations
MAA704:
Matrix
functions and
matrix
equations
Matrix
functions
I
Linear equation system as matrices and column vectors:
Ax = y , A and y known
I
Simple matrix equation: AX = Y or XB = Y , A and y
known or B and y known
I
Sylvester’s equation: AX + XB = C , A, B, C known
I
Liapunov’s equation: AX + XAH = −C , A, C known
Power function
Root function
Matrix
polynomials
Exponential
matrix
Solving matrix
differential
equations
General matrix
functions
Matrix
equations
Tensor products
Matrix
equations,
cont’d
Summary
Solution to Sylvester’s equation
MAA704:
Matrix
functions and
matrix
equations
Matrix
functions
Theorem
Sylvester’s equation AX + XB = C has a unique solution if and
only if A and −B have no common eigenvalues.
Proof.
To show this it is very useful to know about tensor
products.
Power function
Root function
Matrix
polynomials
Exponential
matrix
Solving matrix
differential
equations
General matrix
functions
Matrix
equations
Tensor products
Matrix
equations,
cont’d
Summary
MAA704:
Matrix
functions and
matrix
equations
Tensor products
I
I
Generally: the tensor product on a set is the most general
bilinear function on the set, it is often denoted by ⊗ and
sometimes referred to as the direct product and sometimes
(for certain Hilbert spaces for example) outer product.
Tensor products for matrices always have two properties:
a) Bilinearity:
(µA + ηB) ⊗ C = µA ⊗ C + ηB ⊗ C
Matrix
functions
Power function
Root function
Matrix
polynomials
Exponential
matrix
Solving matrix
differential
equations
General matrix
functions
Matrix
equations
Tensor products
A ⊗ (µB + ηC ) = µA ⊗ B + ηA ⊗ C
b) Associaticity: (A ⊗ B) ⊗ C = A ⊗ (B ⊗ C )
I
For matrices ⊗ is usually the Kroenecker product
Matrix
equations,
cont’d
Summary
MAA704:
Matrix
functions and
matrix
equations
Kroenecker product
Definition
The Kroenecker product ⊗ between two matrices, A ∈ Mm×n
and B ∈ Mp×q , is defined as


a11 B a12 B . . . a1n B
 a21 B a22 B . . . a2n B 


A⊗B = .
..
.. 
..
 ..
.
. 
.
am1 B am2 B . . . amn B
or
Matrix
functions
Power function
Root function
Matrix
polynomials
Exponential
matrix
Solving matrix
differential
equations
General matrix
functions
Matrix
equations

Ab11
 Ab21

A⊗B = .
 ..
Ab12
Ab22
..
.
...
...
..
.

Ab1n
Ab2n 

.. 
. 
Abm1 Abm2 . . . Abmn
these two definitions are equivalent but not equal, one is a
permutation of the other.
Tensor products
Matrix
equations,
cont’d
Summary
Some properties of the Kroenecker product
Theorem
MAA704:
Matrix
functions and
matrix
equations
Matrix
functions
For the Kroenecker product the following is true
a) (µA + ηB) ⊗ C = µA ⊗ C + ηB ⊗ C
b) A ⊗ (µB + ηC ) = µA ⊗ B + ηA ⊗ C
c) (A ⊗ B) ⊗ C = A ⊗ (B ⊗ C )
d) (A ⊗ B)> = A> ⊗ B >
e) A ⊗ B = A ⊗ B (complex conjugate)
H
H
f) (A ⊗ B) = A ⊗ B
H
g) (A ⊗ B)−1 = A−1 ⊗ B −1 , for all invertible A and B
h) det(A ⊗ B) = det(A)k det(B)n , with A ∈ Mn×n , B ∈ Mk×k
Power function
Root function
Matrix
polynomials
Exponential
matrix
Solving matrix
differential
equations
General matrix
functions
Matrix
equations
Tensor products
Matrix
equations,
cont’d
Summary
Some more properties of the Kroenecker product
MAA704:
Matrix
functions and
matrix
equations
Theorem
For the Kroenecker product the following is true
i) (A ⊗ B)(C ⊗ D) = (AC ⊗ BD), A ∈ Mm×n , B ∈ Mn×k ,
C ∈ Mp×q , D ∈ Mq×r
j) A ⊗ B = (A ⊗ Ik×k )(In×n ⊗ B), A ∈ Mn×n , B ∈ Mk×k
e
k) AX = C ⇔ (I ⊗ A)X
>
e
XB = C ⇔ (B ⊗ I )X
where
Power function
Root function
Matrix
polynomials
Exponential
matrix
Solving matrix
differential
equations
General matrix
functions
Matrix
equations
Tensor products

X = X.1 X.2
Matrix
functions

X.1
X.2 

e =
. . . X.n , X
 .. 
 . 
X.n
Matrix
equations,
cont’d
Summary
Eigenvalues and Kroenecker products
MAA704:
Matrix
functions and
matrix
equations
Matrix
functions
Theorem
Let {λ} be the eigenvalues of A and {µ} be the eigenvalues of
B. Then the following is true:
a) {λµ} are the eigenvalues of A ⊗ B
b) {λ + µ} are the eigenvalues of A ⊗ B
Power function
Root function
Matrix
polynomials
Exponential
matrix
Solving matrix
differential
equations
General matrix
functions
Matrix
equations
Tensor products
Matrix
equations,
cont’d
Summary
Eigenvalues and Kroenecker products
Proof.
MAA704:
Matrix
functions and
matrix
equations
Proof (sketch)
a) Use Shurs lemma A = S −1 RA S, B = T −1 RB T where RA
and RB are triangular matrices with eigenvalues along the
diagonal. Then
A ⊗ B = (S −1 RA S) ⊗ (T −1 RB T ) =
= (S −1 RA ⊗ T −1 RB )(S ⊗ T ) =
= (S ⊗ T )−1 (RA ⊗ RB )(S ⊗ T )
Matrix
functions
Power function
Root function
Matrix
polynomials
Exponential
matrix
Solving matrix
differential
equations
General matrix
functions
Matrix
equations
Tensor products
RA ⊗ RB will be triangular and will have the elements
aii bjj = λi µj on the diagonal, thus λi µj is an eigenvalue of
A ⊗ B since similar matrices have the same eigenvalues.
b) Same argument as above gives A ⊗ I (B ⊗ I ) have the same
eigenvalues as A (B) adding the two terms together gives
λ + µ is an eigenvalue.
Matrix
equations,
cont’d
Summary
Solution to Sylvester’s equation, cont’d
MAA704:
Matrix
functions and
matrix
equations
Matrix
functions
Theorem
Sylvester’s equation AX + XB = C has a unique solution if and
only if A and −B have no common eigenvalues.
Proof.
To show this it is very useful to know about tensor
products.
Power function
Root function
Matrix
polynomials
Exponential
matrix
Solving matrix
differential
equations
General matrix
functions
Matrix
equations
Tensor products
Matrix
equations,
cont’d
Summary
Solution to Sylvester’s equation, cont’d
MAA704:
Matrix
functions and
matrix
equations
Matrix
functions
Proof.
Rewrite the equation using the Kroenecker product
e =C
e
AX + XB = C ⇔ (I ⊗ A + B ⊗ I ) X
|
{z
}
K
This is a normal matrix equation which is solvable if λ = 0 is
not an eigenvalue of K .
The eigenvalues of K are λ + µ where λ is an eigenvalue of A
and µ is an eigenvalue of B.
Thus if A and −B have no common eigenvalues the
eigenvalues of K will 6= 0.
Power function
Root function
Matrix
polynomials
Exponential
matrix
Solving matrix
differential
equations
General matrix
functions
Matrix
equations
Tensor products
Matrix
equations,
cont’d
Summary
Summary
MAA704:
Matrix
functions and
matrix
equations
Matrix
functions
I
We
version of common function: An ,
√ have seen matrix
A
A, polynomials, e
I
Seen how analytical functions can be turned into matrix
functions using the Taylor expansion.
I
Introduced the Kroenecker product.
I
Taken a look at some matrix equations, Sylvester’s
equation and first order differential equations.
Power function
Root function
Matrix
polynomials
Exponential
matrix
Solving matrix
differential
equations
General matrix
functions
Matrix
equations
Tensor products
Matrix
equations,
cont’d
Summary