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Chapter 3. Theory of matrices
f
f
M
141
__________________________________________________________________________
and since
Corollary (Rayleigh quotient)
is diagonal matrix, relations (3.5.33) become obvious.
Λ
A
If vectors
b are not normalized (are not of the unit length) formulas (3.5.32) become
⋅⋅
bAb
λ=
max
∈≠
bb
sup
^
()
n
,0
2
b
. (3.5.34)
⋅⋅
bAb
λ=
min
∈≠
bb
inf
^
()
n
,0
2
b
Proof
The proof is analogous to the proof of Proposition 3.5.9.
3.6. Equation Chapter 3 Section 6 Functions
of semisimple matrices
3.6.1. Basic concepts
Herein we will give a formal procedure on how to extend notion of the analytic function of a
scalar argument to the function of a matrix argument, provided such an argument is a square
semisimple matrix.
Definition 3.6.1 (Analytic function of an
Let
function can be represented in a form of the absolutely convergent power series
with scalar coefficients
of a square matrix
substituting matrix
be an analytic function of one variable in a circle
satisfying estimate (1.2.7). Suppose also that all of the eigenvalues
a
k
belong to this circle
∈A
n
in into power series (3.6.1):
A
arbitrary square matrix)
∞
a
k
k
()
xx
=
∑
k
0
=
, (3.6.1)
!
k
, then function of a matrix is defined by
S
of a radius r, then such a
S ∈^
Chapter 3. Theory of matrices
f
L
M
142
__________________________________________________________________________
∞
a
k
Proposition 3.6.1
f
()
=
AA
∑
k
k
k
0
=
. (3.6.2)
!
Assumption that all the eigenvalues belong to the circle of analyticity of function
convergence of the series in the right-hand side of (3.6.2) in any of topologies in the
dimensional vector space of all square matrices.
Proof
Let matrix A has the Jordan normal form:
where
D is a (quasi) diagonal matrix, containing eigenvalues of matrix A and possible
A
Jordan blocks. Substituting representation (3.6.3) in the series (3.6.2) yields:
But each element of matrix
hand side of (3.6.4) is convergent in the strongest topology
Proposition 3.6.2
Any analytic function of a matrix is correctly defined for any
ensures
2
-
n
1−
=⋅⋅
AW D W, (3.6.3)
−
()
AW D W
f
D
A
1
=⋅ ⋅
is contained in a given circle
A
∞
⎛⎞
a
k
k
⎜⎟
∑
⎜⎟
=
k
⎝⎠
A
k
!
0
. (3.6.4)
∞
.
, thus the series in the right
S
nilpotent matrix.
Proof
Indeed, for a nilpotent matrix the series in the right-hand side of (3.6.2) becomes truncated, thus
in such a case there is no need in verification of belonging eigenvalues to the circle of
analyticity.
Proposition 3.6.3 (analytic function of a
If
is a
∈A
n
semisimple matrix, then any analytic function of such a matrix admits a
semisimple matrix)
representation (provided all the eigenvalues of the matrix belong to the circle of analyticity):
()
f
λ
⎛⎞
1
⎜⎟
−
( ) ...
f
AW W
1
=⋅ ⋅
⎜⎟
⎜⎟
⎝⎠
f
()
λ
n
. (3.6.5)
Proof
Let matrix A has the Jordan normal form:
AW WΛ
1−
=⋅⋅
. (3.6.6)
A
Chapter 3. Theory of matrices
143
__________________________________________________________________________
Substituting (3.6.6) into (3.6.2) and recalling that
Remark 3.6.1
From computational point of view formulas (3.6.4), (3.6.5) are much better than (3.6.2).
Proposition 3.6.4
is the diagonal matrix, we arrive at (3.6.5).
Λ
A
Any analytic function
of a square matrix A has the same set of the eigenvectors (and
()f A
possible generalized eigenvectors), as matrix A has.
Proof
The proof flows out from representation (3.6.4) for a non-semisimple matrix and (3.6.5) for a
semisimple matrix.
Corollary
Matrices
A
and
commute.
()f A
3.6.2. Exponent of a matrix
Presumably, the most important function of a matrix found in many applications, is the matrix
exponent defined by (3.6.2):
k
∞
exp( ) ...
AIA
AA
==+++
∑
!2!
k
k
0
=
2
(3.6.7)
Remark 3.6.2
As was pointed out in remark 3.6.1 for computational reasons it is better use formulas (3.6.4),
(3.6.5), than the direct summation in (3.6.7). In this respect for a
Proposition 3.6.5
Matrix exponent is correctly defined for any square matrix A.
Proof
The proof immediately follows from Proposition 3.6.1 and an observation that the exponent
function is analytic everywhere in
exp( )
λ
⎛⎞
⎜⎟
1
exp( ) ...
AW W
−
=⋅ ⋅
⎜⎟
⎜⎟
⎝⎠
^
1
exp( )
λ
.
semisimple matrix
we have
A
. (3.6.8)
n
Chapter 3. Theory of matrices
144
__________________________________________________________________________
Proposition 3.6.6
Suppose that two
semisimple matrices A and B have the same sets of their eigenvectors, then
Proof
Firstly, we remind the in view of Proposition 2.3.4 these matrices commute, and their Jordan
normal forms are
Substituting (3.6.10) into (3.6.7) yields
exp( ) exp( )
But, the exponent of the diagonal matrices satisfies a relation
exp( ) exp( ) exp( )+= ⋅
which is analogous to the well-known relation for the scalar exponent function. Now, from
(3.6.11) and (3.6.12) we get
exp( ) exp( )
+= ⋅ + ⋅=
AB W W
exp( ) exp( ) exp( )+= ⋅AB A B. (3.6.9)
1
−
=⋅⋅
AW W
=⋅⋅
BW W
+= ⋅ + ⋅
AB W WΛΛ
ΛΛ Λ Λ
AB A B
1
−
−
1
=⋅ ⋅ ⋅=
=⋅ ⋅⋅⋅ ⋅=
=⋅
exp( ) exp( )
WW
−−
11
exp( ) exp( )
WWWW
exp( ) exp( )
AB
Λ
A
−
1
Λ
1
−
. (3.6.10)
B
AB
. (3.6.11)
, (3.6.12)
ΛΛ
AB
ΛΛ
AB
ΛΛ
AB
. (3.6.13)
Remark 3.6.3
Assertion of Proposition 3.6.6 for non-semisimple matrices, or semisimple matrices with
different sets of their eigenvectors is generally wrong.
Example 3.6.1
Let symmetric matrix
Its Jordan normal form is
According to (3.6.8)
be
A
21
⎛⎞
=
A
⎜⎟
⎝⎠
1/2 1/2 1 0 1 1
⎛⎞⎛⎞⎛⎞
=⋅⋅
A
⎜⎟⎜⎟⎜⎟
−
1/2 1/2 0 3 1 1
⎝⎠⎝⎠⎝⎠
. (3.6.14)
12
−
. (3.6.15)
Chapter 3. Theory of matrices
145
__________________________________________________________________________
⎛⎞
0
e
1/2 1/2 1 1
exp( )
⎛⎞⎛⎞
=⋅⋅=
A . (3.6.16)
⎜⎟⎜⎟
−
1/2 1/2 1 1
⎝⎠⎝⎠
⎛⎞
⎜⎟
3
0
e
⎝⎠
⎜⎟
−
⎜⎟
⎜⎟
⎜⎟
⎝⎠
33
ee ee
+−+
22
33
−+ +
ee ee
22
This example clearly demonstrate that the exponent of our matrix is not
2
⎛⎞
⎜⎟
⎜⎟
ee
⎝⎠
ee
, (3.6.17)
2
as one, not familiar with the matrix theory, could expect.
3.6.3. Non-integer power of a semisimple matrix
Herein we define power
Since power function
use of power series (3.6.2).
Definition 3.6.2 (Power
Let given semisimple matrix
Then
where power
Λ
α
of a square semisimple matrix for arbitrary real or complex α.
A
α
()fxx
α
of a square
A
α
of diagonal matrix is defined by
A
is not analytic at
=
semisimple matrix for arbitrary α )
A
has the Jordan form
=⋅⋅
AW WΛ
=⋅⋅
AW WΛ
⎛⎞
λ
⎜⎟
α
=
Λ
⎜⎟
A
1−
1α− α
α
1
⎜⎟
⎝⎠
...
, we will give definition for
0x =
. (3.6.18)
A
A
, (3.6.19)
α
without
A
. (3.6.20)
α
λ
n
Remark 3.6.4
For integer α the preceding definition leads to the same result, as the algorithm based on matrix
multiplications.
Example 3.6.2
Applying Definition 3.6.2 for a square root of a given semisimple matrix A yields
Chapter 3. Theory of matrices
146
__________________________________________________________________________
1/2 1 1/ 2−
=± ⋅ ⋅
AW WΛ
A
It can be easily verified that
2
1/2
()
=AA. (3.6.22)
Indeed, powered both sides of (3.6.21) by square, we arrive at (3.6.22).
Let symmetric matrix
1/2
A . (3.6.23)
=⋅⋅=
be as in Example 3.6.1, then its (positive) square root becomes
A
1/2 1/2 1 0 1 1
⎛⎞ ⎛⎞
⎜⎟ ⎜⎟
1/2 1/2 1 1
−
⎝⎠ ⎝⎠
⎛⎞
⎜⎟
⎜⎟
0 3 1313
⎝⎠
−
Again, as it was in Example 3.6.1, power of a matrix (3.6.14) (in our case a square root) is not
just
⎛⎞
21
⎜⎟
⎜⎟
12
⎝⎠
. (3.6.21)
⎛⎞
13 13
+−+
⎜⎟
22
⎜⎟
⎜⎟
−+ +
⎜⎟
22
⎝⎠
, (3.6.24)
as one could expect.
3.6.4. Matrix logarithm of a semisimple matrix
Definition 3.6.3 (Matrix logarithm of a square positive-definite semisimple matrix)
Let given semisimple matrix A has the Jordan form
Then
where
AW WΛ
log log ( )
AW WΛ
()
bb
,
0b >
is the base of logarithm and
1b ≠
log
⎛⎞
⎜⎟
α
=
Λ . (3.6.27)
A
⎜⎟
⎜⎟
⎝⎠
1−
=⋅⋅
−
1
=⋅ ⋅
λ
()
1
b
. (3.6.25)
A
A
log ( )
b A
, (3.6.26)
of diagonal matrix is defined by
Λ
...
log
λ
()
bn
In (3.6.27)
0,..., 0
λ> λ >
1
positive-definiteness of matrix
are the corresponding eigenvalues (positive due to assuming
n
).
A
Chapter 3. Theory of matrices
147
__________________________________________________________________________
Proposition 3.6.7
Suppose that two positive-definite semisimple matrices A and B have the same sets of their
eigenvectors, then
log ( ) log ( ) log ( )
⋅= +AB A B
bbb
. (3.6.28)
Proof
Firstly, we remind the in view of Proposition 2.3.4 these matrices commute, and their Jordan
normal forms are
AW W
=⋅⋅
BW W
Λ
A
−
Λ
. (3.6.29)
B
11−
=⋅⋅
In view of (3.6.29) substituting left-hand side of (3.6.28) into (3.6.26) yields
1
log ( ) log ( )
⋅= ⋅ ⋅ ⋅
AB W WΛΛ
bb
−
AB
. (3.6.30)
But, the logarithm of the diagonal matrices satisfies a relation
⋅= +
log ( ) log ( ) log ( )
ΛΛ Λ Λ, (3.6.31)
bbb
AB A B
which is analogous to the well-known relation for scalar logarithm. Now, from (3.6.30) and
(3.6.31) we get
−
log ( ) log ( )
⋅= ⋅ ⋅ ⋅=
AB W W
bb
1
−
1
=⋅ + ⋅=
WW
−−
11
=⋅ ⋅+⋅ ⋅=
WWWW
log ( ) log ( )
=+
bb
ΛΛ
AB
log ( ) log ( )
ΛΛ
()
bb
AB
log ( ) log ( )
ΛΛ
bb
AB
AB
. (3.6.32)
Proposition 3.6.8
Suppose that matrix A is semisimple and
at
and
0b >
Proof
The proof flows out from Definitions 3.6.2 and 3.6.3.
Proposition 3.6.9
Suppose that matrix
0, 1
bb>≠ commute.
Proof
The proof immediately follows from definition of the matrix logarithm (3.6.26), ensuring
similarity of matrices
positive-definite, then
α
log ( ) log ( )
bb
.
1b ≠
=αAA
(3.6.33)
A is semisimple and positive-definite, then matrices A and
A
and
log ( )
(see Definition 3.5.1).
A
b
log ( )
A
b
at
Chapter 3. Theory of matrices
148
__________________________________________________________________________
Remarks 3.6.5
A.
Assertion of Proposition 3.6.7 for non-semisimple matrices, or semisimple matrices with
different sets of their eigenvectors is generally wrong.
Matrix logarithm serves as an inverse function to the matrix exponent. Indeed, according to
B.
(3.6.8) and Definition 3.6.3
1
−
ln exp( ) ln exp
()
=⋅⋅=
AWW
ln exp( )
WW
()
()
(
1
−
WW
()
(
1
−
ln exp( )
⋅⋅
Λ
A
Λ
⋅⋅=
()
Λ
A
A
)
. (3.6.34)
)
However, while
positive-definite (and semisimple) square matrices.
exp( )A
is well defined for any square matrix A,
3.6.5. Trigonometric and hyperbolic functions
of a square matrix
Definition 3.6.4 (Trigonometric functions of a square semisimple matrix)
Let given semisimple matrix A has the Jordan form
Then, according to the general rule given in Proposition 3.6.3
AW WΛ
sin sin( )
AW W
()
cos cos( )
AW W
()
tan tan( )
AW W
()
cot cot( )
AW W
()
1−
=⋅⋅
1
−
=⋅ ⋅
1
−
=⋅ ⋅
1
−
=⋅ ⋅
1
−
=⋅ ⋅
. (3.6.35)
A
Λ
A
Λ
Λ
Λ
A
A
A
, (3.6.36)
log ( )
A
b
is defined for
where
condition
Definition 3.6.5 (Hyperbolic functions of a square (not necessary semisimple) matrix)
Let given semisimple matrix A has the Jordan form
Then, according to the general rule given in Proposition 3.6.3
tan( )
,1,...,
knλ=
k
is well defined, provided eigenvalues
Λ
A
/2 ,
k
satisfy condition
ppλ≠π +π ∈]
. Similarly,
,
ppλ≠π ∈]
k
=⋅⋅
AW WΛ
cot A
1−
A
,1,...,
knλ=
k
is well defined, provided eigenvalues
()
.
. (3.6.37)
of matrix
satisfy
A
Chapter 3. Theory of matrices
149
__________________________________________________________________________
1
sinh sinh( )
AW W
()
cosh cosh( )
AW W
()
tanh tanh( )
AW W
()
coth coth( )
AW W
()
−
=⋅ ⋅
1
−
=⋅ ⋅
1
−
=⋅ ⋅
1
−
=⋅ ⋅
Λ
Λ
Λ
Λ
A
A
A
A
, (3.6.38)
where
tanh( )
provided eigenvalues
Proposition 3.6.10
Suppose that square matrix A satisfy condition of existence for corresponding trigonometric or
hyperbolic function, then matrices
function, commute.
Proof
The proof flows out directly from Definitions 3.6.4 and 3.6.5.
Remark 3.6.5
It is interesting to note that there exist relations between some of matrix trigonometric and
hyperbolic functions resembling ones existing between corresponding scalar functions. For
example,
is well defined for any square matrix. Similarly,
Λ
A
,1,...,
knλ=
k
sin ( ) cos ( )
sin(2 ) 2sin( ) cos( )
cos(2 ) cos ( ) sin ( )
do not vanish:
and
A
22
AAI
+=
AAA
=⋅
AAA
22
=−
()f A
0
λ≠
k
, where f stands for the corresponding
. (3.6.39)
.
coth A
()
is well defined,
Similarly,
22
sinh ( ) cosh ( ) exp( )
sinh ( ) cosh ( ) exp( )
exp( ) cos( ) sin( )
AAA
+=
22
AA A
−=−−
AA A
=+
ii
. (3.6.40)
However, relations involving different matrices, for example
sin( ) sin( ) cos( ) cos( ) sin( )
+= ⋅ + ⋅
remain valid, if matrices
AB A B A B
+= ⋅ − ⋅
AB A B A B
cos( ) cos( ) cos( ) sin( ) sin( )
and B are similar.
A
, (3.6.41)
Chapter 3. Theory of matrices
M
M
150
__________________________________________________________________________
3.7. Equation Chapter 3 Section 7 Functions
of non-semisimple matrices
3.7.1. Basic concept
Herein we will give a formal procedure on how to extend notion of the analytic function of a
scalar argument to the function of a matrix argument, provided such an argument is a square
non-semisimple matrix.
Definition 3.7.1 (Analytic function of a non-semisimple matrix)
Analytic function of an arbitrary non-semisimple matrix can be defined by substituting a nonsemisimple matrix into Taylor series (3.6.1).
Remark 3.7.1
However, our aim is to define an analytic function of a non-semisimple matrix in a way that is
better suited for applications. Let a non-semisimple matrix
is represented by its Jordan
∈A
n
normal form:
where
matrix
A
blocks of
is a matrix composed of all left eigen- and the generalized eigenvectors of
∈W
n
, and D is a block-diagonal matrix containing diagonal elements and all the Jordan
A :
λ
⎛⎞
1
⎜⎟
⎜⎟
⎜⎟
⎜⎟
D
=
-1
= ⋅⋅AW DW
...
λ
p
⎜⎟
⎜⎟
⎜⎟
⎜⎟
⎝⎠
In (3.7.2)
denotes the Jordan block of the index
J
k
m
of determining function of a matrix for a
arguments, define a procedure for
non-semisimple matrices:
, (3.7.1)
J
k
1
. (3.7.2)
...
J
k
q
. Since Proposition 3.6.3 gives the rule
k
m
semisimple matrix, we now by applying analogous
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