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Файл:Additional Chapters of Higher Mathematics for Masters in Civill and Geotechnical Engineering. Учебное пособие по дополнительным разделам высшей матема
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Chapter 3. Theory of matrices
M
111
__________________________________________________________________________
23 45
ii
++
then
Definition 3.1.6 (Kernel space of a matrix)
⎛⎞
=
A
⎜⎟
++
ii
78 16
⎝⎠
ii
−−
23 78
⎛⎞
=
A
*
⎜⎟
−−
ii
45 16
⎝⎠
, (3.1.12)
(3.1.13)
Right kernel space of a matrix
-dimensional vectors m satisfying a condition:
m
is a vector subspace (possibly empty) consisted of all
M∈A
,nm
. (3.1.14)
0⋅=Am
Left kernel space consists of all n -dimensional vectors m satisfying condition:
. (3.1.15)
0⋅=mA
Kernel space (without indicating left or right) will be referred to the right kernel space. This
space will be denoted by ker( )
A .
3.1.3. Determinants
Definition 3.1.7 (Permutation and transposition)
Let
permutations. Permutations that alter only two elements of the set
transpositions. It is easy to show that
represented as a product (successive application) of different transpositions. According to the
number of transpositions needed to get a given permutation, the permutation is called
odd. The sign of permutation is defined by the following condition:
be a group of all one-to-one mappings of the set
S
n
contains !n elements. Each permutation
S
n
[1,..., ]n
. These mappings are called
[1,..., ]n
are called
can be
Sσ∈
n
even or
Definition 3.1.8 (Determinant)
Let
be a square matrix. Determinant
∈A
n
the following formula:
1,
if is even
+σ
sgn
det( ) sgn
⎧
σ=
⎨
−σ
1,
if is odd
⎩
=σ
A
∑ ∏
Si
σ∈ =
n
det( )A
n
1
. (3.1.16)
of the matrix A is a number, defined by
. (3.1.17)
a
ii
()
σ
Chapter 3. Theory of matrices
M
M
M
112
__________________________________________________________________________
Thus, the determinant is the sum of
of the matrix.
elements, each being a product of n components
!n
a
()ii
σ
Example 3.1.2 (Determinant of a square
According to Definition 3.1.7, group
22×
S
(1)
:[1,2] [1,2]
σ→
(2)
σ→
Now, applying (3.1.17), (3.1.18), the determinant of a square
2
det( ) sgn
=σ =
A
∑
k
=
sgn sgn
()
k
aa
() ()
kk
(
(1)1 ( 2) 2
1
(1) (2 )
σ+σ =−
()
σσ
aa aa aa a a
()
11 22 21 12 11 22 21 12
Proposition 3.1.4 (Properties of determinants)
det( ) det( )det( )⋅=AB A B
det( ) 1=I
det( ) 1
-matrix)
contains two elements:
2
. (3.1.18)
:[1,2] [2,1]
)
()
⋅=AA
()
, (3.1.20)
, (3.1.21)
1
−
, (3.1.22)
22×
matrix
is
∈A
2
(3.1.19)
det( ) det( )
ccc
∀=AA
c
∈
^
Corollary
The determinant retains the multiplicative structure of
Proposition 3.1.5 (Determinant of a block matrix)
Let
where
be a block matrix:
R
,,,
∈ABCD
det( ) det( )det( ), if det( ) 0
det( ) det( ) det( ), if det( ) 0
det( ) det( ) det( ), if det( ) 0
AB
⎛⎞
=
R
⎜⎟
CD
⎝⎠
, then
n
=−⋅⋅ ≠RADCAB A (3.1.25)
=−⋅⋅ ≠RDABDC D (3.1.26)
=−⋅⋅ ≠RDABDC D (3.1.27)
n
. (3.1.23)
.
n
. (3.1.24)
1
−
1
−
1
−
Chapter 3. Theory of matrices
113
__________________________________________________________________________
1
det( ) det( ) det( ), if det( ) 0
=−⋅⋅ ≠RDABDC D (3.1.28)
Proof
To prove (3.1.25) we multiply (from left) the first raw in (3.1.24) by
raw from the second one, this yields:
AB
⎛⎞
=
R
⎜⎟
1
⎜⎟
0DCA B
⎝⎠
−
1
−⋅ ⋅
−
1−
⋅CA and subtract this
. (3.1.29)
The determinant of a newly constructed matrix
that is because we applied only linear transformations to construct
R coincides with the determinant of matrix R ,
1
R
. But
1
det( )R
can easily be
1
constructed from (3.1.29), which gives (3.1.25). Proofs of Eqs. (3.1.26) – (3.1.28) are analogous.
Remark 3.1.1
Proposition 3.1.5 admits generalization to determinants of higher ranks. For example, the
determinant of a following
composed of square matrices
RA
det( ) det( )det
=
11
In obtaining (3.1.31) it is assumed that matrix
-block matrix:
33×
AAA
⎛⎞
11 12 13
⎜⎟
RA A A
=
21 22 23
⎜⎟
⎜⎟
AAA
31 32 33
⎝⎠
, can be represented in a form that resembles (3.1.25):
A
ij
11
⎛⎞
−⋅⋅ −⋅⋅
A AAA A AAA
22 21 11 12 23 21 11 13
⎜⎟
AAAAAAAA
−⋅⋅ −⋅⋅
32 31 11 12 33 31 11 13
⎝⎠
−−
11
−−
A
(3.1.30)
. (3.1.31)
is not degenerate. The last determinant in the
11
right-hand side of (3.1.31) can be computed by one of the formulas (3.1.25) - (3.1.28).
It should also be noted that the algorithm in obtaining (3.1.31) is similar to what is known as
“the reduction of a square matrix to the upper triangular matrix”.
Chapter 3. Theory of matrices
M
R
114
__________________________________________________________________________
3.2. Equation Chapter 3 Section 2
Eigenproblems
3.2.1. Eigenvalues and eigenvectors
Definition 3.2.1 (Eigenvalue and eigenvector)
Let
eigenvalue λ, if:
Similarly, vector
The complete set of all eigenvalues of matrix
Remark 3.2.1
A.
An eigenvalue can take any value in ^ , while an eigenvector is necessary non-zero and
B.
It will be shown in the next section that left
be a square matrix. Vector m is called the right eigenvector corresponding to the
∈A
n
⋅=λAm m
m is called the left eigenvector, if
⋅=λmA m
generally complex vector. Expressions (3.2.1), (3.2.2) show that eigenvectors are
defined up to scalar multipliers.
corresponding to different eigenvalues
condition:
LR
⋅=mm
12
. (3.2.1)
. (3.2.2)
A is called spectrum and is denoted by Sp( )A .
L
and right
m
1
and
λ
1
. (3.2.3)
0
, may obey the orthogonality
λ
2
eigenvectors
m
2
Proposition 3.2.1
Equations (3.2.1), (3.2.2) admit the following equivalent forms:
and
Corollary 1 (Characteristic polynomial)
All eigenvalues satisfy the following (polynomial with respect to
characteristic polynomial equation:
(3.2.4)
()
()
0−λ ⋅ =AIm
. (3.2.5)
0⋅−λ=mA I
) equation, known as the
λ
Chapter 3. Theory of matrices
M
M
M
M
115
__________________________________________________________________________
() det 0ψλ ≡ −λ =AI
Corollary 2 (Theorem on power of the spectrum)
()
. (3.2.6)
Any square matrix
multiple). Thus, spectrum Sp( )A of any square matrix
Proof
The proof flows out from Eq. (3.2.6), which is a polynomial equation of the order n with
respect λ .
Remark 3.2.2
An arbitrary
matrix
has the following characteristic polynomial
which has only imaginary roots:
has exactly
∈A
n
real square matrix
() det 1
ψλ = =λ +
, generally complex, eigenvalues (some may be
n
may have no
∈A
n
01
⎛⎞
=
A
⎜⎟
−
⎝⎠
−λ
⎛⎞
⎜⎟
1
−−λ
⎝⎠
,iiλ=+ λ=−
12
(3.2.7)
10
1
2
. (3.2.9)
contains exactly
∈A
n
n elements.
real eigenvalues at all. For example, real
, (3.2.8)
Proposition 3.2.2
Right (left) eigenvectors corresponding to different eigenvalues, are linearly independent.
Proof
Let
λ≠λ
12
corresponding right eigenvectors. Suppose that these eigenvectors are linearly dependent.
Multiplying one of the eigenvectors by a non-vanishing constant, we can assume that
Applying (3.2.1), we arrive at
3.2.2. Properties of eigenvectors
be two different eigenvalues of the matrix
(3.2.10)
=mm
12
()−=λ−λAm m m m
12 1122
(3.2.11)
, and
∈A
n
be the
,mm
12
Chapter 3. Theory of matrices
A
M
M
x
M
116
__________________________________________________________________________
Now, in view of (3.2.10), the left-hand side of Eq. (3.2.11) vanishes, while the right-hand side
does not, since
corresponding to
λ≠λ
different eigenvalues is proved analogously.
k
Definition 3.2.2 (Algebraic multiplicity; Geometric multiplicity)
Algebraic multiplicity of an eigenvalue is the multiplicity of the corresponding root of
A.
the characteristic polynomial.
B.
Geometric multiplicity of an eigenvalue is the number of linearly independent right (left)
eigenvectors, corresponding to this eigenvalue.
Proposition 3.2.3
by assumption. A more general case of
12
eigenvectors
1k >
( )( )
lgebraic multiplicity Geometric multiplicity≥ (3.2.12)
Proof
Let
3.2.2 there are
and m be the geometric multiplicity of an eigenvalue. According to Definition
∈A
n
linearly independent eigenvectors forming some m-dimensional vector
m
space. Restriction of matrix
Corollary 2 of Proposition 3.2.1 ensuring existence of
is proved.
3.2.3. Properties of characteristic and minimal
Definition 3.2.3 (Minimal polynomial)
Let
of the lowest order, for which
, a normalized polynomial
∈A
n
() ...
ϕ≡ + ++
A on this m -dimensional vector space allows us to apply
m eigenvalues. Thus, inequality (3.2.12)
polynomials
mm
xax a
1
−
1
(3.2.13)
m
() ... 0
ϕ≡ + ++ =AA A I
is called the minimal polynomial.
Theorem 3.2.1 (Cayley – Hamilton)
If
is a characteristic polynomial of a given square matrix
ψ
mm
aa
1
−
1
m
, (3.2.14)
, then
∈A
n
Chapter 3. Theory of matrices
M
I
I
117
__________________________________________________________________________
Proof
The proof immediately follows from (3.2.6) by substituting matrix A instead of λ.
Corollary 1
Let ϕ and ψ be minimal and characteristic polynomials respectively, then
() 0ψ=A . (3.2.15)
Proof
The proof is obvious, since ϕ is the minimal polynomial (of the lowest degree).
Corollary 2
Minimal polynomial ϕ divides characteristic polynomial ψ .
Proof
In view of (3.2.16), we can write
() ()() ()
where ,
ch are some polynomials, and deg( ) deg( )h <ϕ, since h is the residual. Combining
(3.2.14), (3.2.15), we get from (3.2.17):
Thus,
is the divisor of ψ.
ϕ
() ()() () 0 () 0ch hψ= ϕ+ =⇒ =AAAA A
deg( ) deg( )ϕ≤ ψ. (3.2.16)
chψλ = λϕλ + λ , (3.2.17)
. (3.2.18)
Herein, the exposition of matrix invariants follows Ericksen (1960).
Definition 3.2.4 (Matrix invariant)
k -th (principle) invariant of a matrix
The
(some) of the elements of matrix
polynomial:
ψλ=λ − λ + + −
In the right-hand side of (3.2.19) the matrix invariants are denoted by
3.2.4. Matrix invariants
is the k -th order symmetric function of
∈A
n
, coinciding with coefficients of the characteristic
A
1
() ( )
nn
−
1
... 1
n
(3.2.19)
I
n
1
,...,
.
I
n
Chapter 3. Theory of matrices
M
M
∑
I
I
118
__________________________________________________________________________
Proposition 3.2.4
Matrix invariants of a matrix
remain invariant at arbitrary non-degenerate
∈A
n
transformations.
Proof
Considering expression (3.2.6) for the characteristic polynomial of matrix
matrix
()
from both sides by non-degenerate matrices W and
−λAI
11
det det det
−λ = ⋅ −λ ⋅ = ⋅ ⋅ −λAI WAIW WAW I
() ()
()()
−−
Expressions (3.2.20) complete the proof.
Corollary
Matrix invariants of a given square matrix
coincide with invariants of its Jordan
∈A
n
normal form (see Sections 3.3 and 3.4 for the corresponding definitions for the Jordan normal
forms).
Remarks 3.2.3
A. Combining expressions (3.2.6) and (3.2.19) gives the following simple expressions for first
and last invariants
A and multiplying
1−
W , yields
. (3.2.20)
Ia a a=+ ++ ≡A
11122
... tr( )
det( )
I = A
n
nn
. (3.2.22)
. (3.2.21)
B. According to Corollary to Proposition 3.2.4 the first and last invariants of a matrix
easily expressed in terms of its eigenvalues:
n
C. When 3
n = the principle invariants of a matrix (or a second-order tensor) A are often
denoted by
, and
II
,
AA
I
=λ
1
=λ
nk
. In such a case the second invariant admits a relatively
II
A
, (3.2.23)
k
k
1
=
n
∏
kI=
. (3.2.24)
1
simple representation
1
2
II
A
In terms of the eigenvalues of matrix
tr ( ) tr( )
=−⋅=
2
aa a a aa a a a
11 22 22 33 33 11 12 23 31
AAA
()
++−−−
the second invariant takes the form:
A
222
. (3.2.25)
can be
A
II =λ λ +λ λ +λ λ
12 23 31
A
. (3.2.26)
Chapter 3. Theory of matrices
119
__________________________________________________________________________
The second invariant of a square 3 3× or 2 2× matrix (or better speaking of a second-order
tensor) plays a very important role in mechanics of deformable solids. Quite often a square
root of the second invariant of an auxiliary matrix is associated with the overall stress, or
more precisely:
where matrix
1
tr( )−AAI
3
eigenvalues of matrix
1/2 2
(3)
σ= λ −λλ −λλ −λλ =
= λ −λ + λ −λ + λ −λ
In mechanics
σ is sometimes called as v.Mises stress.
There are several (equivalent) norms suggested for matrices
Definition 3.2.5 (Frobenius norm)
Let
M∈A ^
nm
be arbitrary complex matrix. The Frobenius norm is defined by:
()
,
II
−σ=AAI
, (3.2.27)
1
tr( )
3
is known as the deviator of matrix
A the overall stress
3
∑
k
k−=
1
−
(6) ( ) ( ) ( )
12 23 31
1/2 2 2 2
12 23 31
admits the following representation:
σ
3.2.5. Matrix norms
. In terms of the
A
. (3.2.28)
≡⋅⋅= =
AAA
F
*
nm nm
∑∑ ∑∑
11 11
kl kl
== ==
Remarks 3.2.4 (Hilbert-Schmidt norm; Schur norm)
A.
In case of square matrices Frobenius norm is sometimes called Hilbert-Schmidt or Schur
norm.
B.
If
M∈A ^
is a square matrix and
()
n
then Frobenius norm coincides with
n
≡λ
A
F
∑
k=
It can be shown (see Gantmaher, 2005) that formula (3.2.30) remains valid for any square
matrix
M∈A ^
regardless of its properties (degeneracy, simplicity, non-
()
n
semisimplicity, etc.).
aa a
kl kl kl
, 1,...,
knλ=
k
2
k
1
. (3.2.30)
are the corresponding eigenvalues,
2
. (3.2.29)
Chapter 3. Theory of matrices
p
p
p
p
I
120
__________________________________________________________________________
Definition 3.2.6 (Frobenius
-norm)
Let
M∈A ^
nm
be arbitrary complex matrix. The Frobenius p-norm generalizes the
()
,
previous definition:
A
,
Fp
≡
Remarks 3.2.5
A. At
B.
However, in contrast to the case 2p = , for which relation (3.2.30) holds, at other values of
norm (3.2.31) transforms into Frobenius norm and at
2p =
,
F
no such a relation exists.
Definition 3.2.7 (Nuclear norm; Ky Fan norm)
Let
M∈A ^
nm
be arbitrary complex matrix. The nuclear norm is defined by:
()
,
≡⋅
AAA
*
nm
⎛⎞
⎜⎟
∑∑
⎜⎟
11
==
kl
⎝⎠
max
≡A . (3.2.32)
∞
,
kl
⎛⎞
tr
⎜⎟
⎝⎠
1/
p
a
kl
a
kl
*
. (3.2.31)
we arrive at
=∞
. (3.2.33)
Remarks 3.2.6
A. Matrix
matrix is well defined (see Sec.3.6).
B.
If A is a square matrix, then it is obvious that the nuclear norm coincides with the first
invariant
and representation for the square root of a square matrix in terms of its Jordan normal form
(Sections 3.6 and 3.7), yields
for any square matrix
Example 3.2.1
Herein we consider computation of different matrix norms for the following matrix:
*
is a square (and positive semi definite matrix), so the square root of such a
⋅AA
of the matrix
1
A
.
*
. Taking into account Corollary from Proposition 3.2.4
⋅AA
=AA (3.2.34)
* F
i
−−
01
⎛⎞
=
A
⎜⎟
i
10
⎝⎠
(3.2.35)
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