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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
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where
B.
Characteristic (
the index
is the nilpotent matrix of the n -th index of nilpotency.
N
n
) and minimal (ϕ) polynomials for the nilpotent matrix
ψ
coincide and have the form:
n
∈N
nn
of
Characteristic ( ψ ) and minimal ( ϕ ) polynomials for the Jordan block
C.
() ()
n
xxψ=ϕ=
. (3.4.13)
coincide and have the form:
n
Both the Jordan block
D.
the same unique
The corresponding
E.
The generalized eigenvectors
nilpotent matrix
() () ( )
xxxψ=ϕ=−λ
∈J and the corresponded nilpotent matrix
nn
right eigenvector:
1
⎛⎞
⎜⎟
0
⎜⎟
=
m . (3.4.15)
⎜⎟
...
⎜⎟
0
⎝⎠
left eigenvector is a vector obtained from (3.4.15) by transposing.
of the index p for the Jordan block
m
are the same. According to their indices, the corresponding
∈N
nn
. (3.4.14)
generalized eigenvectors are:
00
⎛⎞ ⎛⎞
⎜⎟ ⎜⎟
10
⎜⎟ ⎜⎟
⎜⎟ ⎜⎟
0 , ... , ...
==
mm
2
⎜⎟ ⎜⎟
... 0
⎜⎟ ⎜⎟
⎜⎟ ⎜⎟
01
⎝⎠ ⎝⎠
n
. (3.4.16)
∈J
nn
∈N have
nn
∈J and the
nn
Proofs
Proof A flows out from definitions for the Jordan block (3.4.1) and nilpotent matrix. Proofs B
and C follow from representation (3.4.12) and definition for the nilpotent matrix. Proofs D and
E are obtained by direct verification.
Remark 3.4.2
It follows from (3.4.15), (3.4.16) that the Jordan blocks and the corresponding nilpotent
matrices in
(including one genuine eigenvector).
have the same complete set of n mutually orthogonal generalized eigenvectors
n
Chapter 3. Theory of matrices
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Proposition 3.4.2 (
Jordan normal form of a general matrix)
Any matrix
by a non-degenerate transformation
∈A
n
can be reduced to the (block)
∈W
n
diagonal form:
1−
=⋅⋅AW DW, (3.4.17)
where
diag( ,..., , ,..., )
is a block-diagonal matrix containing scalars
,...,
scalars
λλ are the corresponding eigenvalues, while numbers in brackets indicate ordinal
1
k
numbers, and not ranks of the Jordan blocks. Columns of matrix
=λλDJJ (3.4.18)
1(1)()
kp
and Jordan blocks
,...,
λλ
1
k
,...,
JJ
(1) ( )
1−
W are either (right)
eigenvectors or the generalized eigenvectors associated with the corresponding Jordan blocks.
Proof
Proof is similar to proof of Proposition 2.3.3.
Remark 3.4.3
If
is a non-semisimple invertible matrix, then the analogue of Proposition 2.3.5 is not
∈A
n
valid. A non-semisimple matrix and its inverse may have different sets of the (generalized)
eigenvectors.
. In (3.4.18)
Proposition 3.4.3
Any two Jordan blocks of the same rank and the same index commute (index of the Jordan
block is the index of nilpotency of the related nilpotent matrix).
Proof
Let
J
and
1
where index
Direct multiplication yields
Example 3.4.3
The following example illustrates the preceding remark. Let
Jordan block of the third rank:
be two Jordan blocks of the following structure:
J
2
=λ + =λ +JINJ IN
()( )
11 2 2
denotes the index of nilpotency of the corresponding nilpotent matrix.
k
⋅=λλ+λ+λ + =⋅JJ I N N J J
12 12 1 2 1 21
()
,
kk
()
kk−
, (3.4.19)
. (3.4.20)
be the non-degenerate
∈J
3
Chapter 3. Theory of matrices
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510
⎛⎞
⎜⎟
=
J
051
⎜⎟
⎜⎟
005
⎝⎠
The eigenvector and two generalized eigenvectors form the corresponding eigenmatrix:
100
⎛⎞
⎜⎟
=
W
010
J
⎜⎟
⎜⎟
001
⎝⎠
At the same time, the inverse to J is
111
⎛⎞
−
⎜⎟
5 25 125
⎜⎟
1
−
⎜⎟
=−
J
⎜⎟
11
0
525
⎜⎟
⎜⎟
00
⎜⎟
⎝⎠
. (3.4.21)
. (3.4.22)
. (3.4.23)
1
5
But the eigenmatrix for
stored columnwise, is
Thus, matrices
Remark 3.4.4
Generally, reduction of a particular matrix
possibility to conclude, whether such a matrix is semisimple or not. The following example is
concerned with semisimple and non-semisimple matrices.
Example 3.4.4
W and
J
1−
with the genuine eigenvector and the generalized eigenvectors
J
1
W
are different.
W
1−
J
⎛⎞
⎜⎟
625
⎜⎟
⎜⎟
0
=−
1
−
J
⎜⎟
⎜⎟
001
⎜⎟
⎜⎟
⎝⎠
00
11
. (3.4.24)
25 5
to the Jordan normal form is the only
∈A
n
The following matrix
which is
⎛⎞
=
A
⎜⎟
−
i
⎝⎠
complex-symmetric one, is not semisimple, as being reduced to the Jordan normal
, (3.4.25)
3
i
−
5
form, it leads to appearing the Jordan block of the second rank:
Chapter 3. Theory of matrices
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11 41 0
But a slight modification of matrix (3.4.25)
⎛⎞⎛⎞⎛⎞
=⋅⋅
A
⎜⎟⎜⎟⎜⎟
−−
ii
0041
⎝⎠⎝⎠⎝⎠
5
⎛⎞
=
B
⎜⎟
−
⎝⎠
i
−
i
4
i
. (3.4.26)
(3.4.27)
gives a semisimple matrix (the latter is also a
normal form is:
⎛⎞
ii
−+ +
33 33
⎜⎟
66
⎜⎟
B (3.4.28)
This example demonstrates practical useless of notion of the symmetry applied to the complex
matrices.
=⋅⋅
⎜⎟
⎜⎟
⎝⎠
33 93 3
−−
33 2 2
⎛⎞⎛⎞
++
93 3
⎜⎟⎜⎟
⎜⎟⎜⎟
⎜⎟⎜⎟
⎜⎟⎜⎟
⎝⎠⎝⎠
complex-symmetric matrix). Indeed, its Jordan
ii
22
01
01
ii
−−
−
3.5. Equation Chapter 3 Section 5 Matrix
classes
3.5.1. Basic matrix classes
Definition 3.5.1 (Similar matrices)
Two square matrices
such that
Remark 3.5.1
In view of Propositions 2.3.3 and 2.4.2 similar matrices can be reduced to the Jordan normal
form by the same non-degenerate transformation matrix
,
are called similar if there is a non-degenerate matrix
∈AB
n
1−
=⋅⋅AW BW (3.5.1)
W .
,
∈W
n
Chapter 3. Theory of matrices
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Definition 3.5.2 (Normal matrix)
Matrix
is called a normal matrix, if
∈A
n
**⋅=⋅AA A A. (3.5.2)
See Definition 3.1.5 for the complex conjugate matrix.
Definition 3.5.3 (Hermitian matrix)
Matrix
is called a Hermitian matrix, if
∈A
n
Definition 3.5.4 (Unitary matrix)
Matrix
is called a unitary matrix, if
∈A
n
Remark 3.5.2
It can be shown that relation (3.5.4) ensures rows and columns of the unitary matrix to be
orthonormal (orthogonal and normal).
. (3.5.3)
*=AA
1
−
*
=AA. (3.5.4)
Definition 3.5.5
Matrix
∈A is called a skew-Hermitian matrix, if
n
In the case matrix
Definition 3.5.6
Matrix
(Normal real matrix)
∈A
n
is real the following definitions are applicable:
∈A
n
is called a normal matrix, if
Definition 3.5.7 (Symmetric matrix)
Matrix
is called a symmetric matrix, if
∈A
n
* =−AA
tt
⋅=⋅AA A A. (3.5.6)
. (3.5.5)
t
. (3.5.7)
=AA
Chapter 3. Theory of matrices
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Definition 3.5.8 (Orthogonal matrix)
Matrix
is called an orthogonal matrix, if
∈A
n
Remark 3.5.3
It can be shown that relation (3.5.8) ensures rows and columns of the orthogonal matrix to be
orthonormal (orthogonal and normal).
Definition 3.5.9 (Skew-symmetric matrix)
Matrix
is called a skew-symmetric matrix, if
∈A
n
Remark 3.5.4
We will not be using symmetry of complex matrices, as it is practically useless (see Example
2.4.3), regarding symmetry of the real matrices only.
1 t−
=−AA
. (3.5.8)
=AA
t
. (3.5.9)
3.5.2. Relations between matrix classes, eigenvectors,
Proposition 3.5.1
Similar matrices, having Jordan blocks of the same rank (at reducing to the Jordan normal
form), commute.
Proof
For semisimple matrices this is due to Proposition 2.3.4, for non-semisimple matrices the proof
follows from an observation that the Jordan blocks of the same rank, commute.
Proposition 3.5.2
If real matrices
Proof
The desired relation
flows out from normality and commutativity of the given matrices
and eigenvalues
,
are normal and commute, then matrix
∈AB
n
⋅⋅⋅ = ⋅ ⋅⋅AB AB AB AB (3.5.10)
()()()()
tt
is a normal matrix.
⋅
AB
and B.
A
Chapter 3. Theory of matrices
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Proposition 3.5.3
Hermitian, unitary, skew-Hermitian, symmetric, orthogonal, and skew-symmetric matrices are
normal.
Proof
The proof is obvious.
Proposition 3.5.4
Proof
These implications immediately follow from the corresponding definitions.
Proposition 3.5.5
A.
All the eigenvalues of a Hermitian matrix are real
All the eigenvalues of a symmetric matrix are real
B.
Absolute values of eigenvalues of an unitary matrix are unit
C.
Absolute values of eigenvalues of an orthogonal matrix are unit
D.
Proof
To prove Condition A, it is sufficient to observe that if A is a Hermitian matrix,
and
m is the corresponding (generally complex) eigenvector with
⎛⎞
emisimple normal skew Hermitian skew symmetric
⇒⇒
⎜⎟
⎜⎟
⎜⎟
⎝⎠
Hermitian symmetric
− ⇒ −
unitary orthogonal
()
⇒
⇒
1=m
. (3.5.12)
0⋅−λ⋅=mA Im
(3.5.11)
Sp( )λ∈ A
, then
,
Equation (3.5.12) implies
Taking complex conjugate of both sides of (3.5.13) implies
But since
. Proof of Condition B is analogous.
λ=λ
To prove C we observe that for the unitary matrix
At the same time, due to Proposition 2.3.5 we have for the inverse matrix:
Combining (3.5.15), (3.5.16) and taking into account that
is Hermitian, left-hand side of (3.5.13) coincides with (3.5.14). This ensures that
A
⋅⋅ =λmAm
⋅⋅ =λmAm
*⋅⋅=λmU m
11−−
⋅⋅=λmU m . (3.5.16)
. (3.5.13)
. (3.5.14)
condition (3.5.14) implies
U
. (3.5.15)
U is a unitary matrix, we arrive at
Chapter 3. Theory of matrices
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Equation (3.5.17) completes the proof. Proof of Condition D is analogous.
Proposition 3.5.6
Any normal matrix possesses the complete set of orthonormal eigenvectors.
Corollary
Any normal matrix can be reduced to the diagonal form by applying an orthogonal or unitary
transformation.
Remark 3.5.5
We have already seen (Corollary 2 to Proposition 2.3.2) that any semisimple matrix has the
complete set of linearly independent eigenvectors and the fundamental matrices composed of its
left and right eigenvectors are mutually orthogonal. Proposition 3.5.6 ensures existence of the
orthonormal set of left (or right) eigenvectors for any
λ=λ . (3.5.17)
1−
normal matrix.
Proposition 3.5.7
A. Matrix is Hermitian, iff it can be reduced to the Jordan normal form by a unitary
transformation, and the corresponding diagonal matrix contains only real numbers.
Matrix is symmetric, iff it can be reduced to the Jordan normal form by an orthogonal
B.
transformation, and the corresponding diagonal matrix contains only real numbers.
Proof
Let matrix
we get:
but since unitary matrix is not degenerate, (3.5.18) means
And, as Λ is a diagonal matrix, (3.5.19) yields Im( )=0Λ . The inverse implication flows out
directly from (3.5.18) and (3.5.19). The proof of condition B is analogous.
3.5.3. Positive definiteness and related problems
be Hermitian, then reducing both sides of (3.5.3) to the Jordan normal form
∈A
n
UUU UΛΛ
***⋅⋅ = ⋅ ⋅
. (3.5.19)
*=ΛΛ
, (3.5.18)
Definition 3.5.10 (Positive definite matrix)
Matrix
is called positive definite, if
∈A
n
Chapter 3. Theory of matrices
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∀⋅⋅>
bbAb
n
,0
∈≠
bb
^
Condition (3.5.20) actually assumes that
(3.5.20)
0
Remark 3.5.6
symmetric matrices vectors b in the preceding definition may belong to the real vector
For
n
space
\
.
Proposition 3.5.8
If
∈A
n
Condition (3.5.22) means that all the eigenvalues should be real and positive.
Proof
Let given Hermitian matrix A have the Jordan normal form:
*=⋅⋅AU UΛ . (3.5.23)
Since unitary matrix U is not degenerate, it defines a one to one mapping of the vector space
n
onto itself. So, an arbitrary non-vanishing vector
^
∀⋅⋅=
n
∈≠
bb
^
Hermitian or symmetric matrix, then condition (3.5.20) is equivalent to
is a
Im 0
bbAb
,0
()
Sp( )
. (3.5.22)
⊂A \
+
. (3.5.21)
n
can be chosen in a form:
∈b ^
n
where '
∈b ^ is some also non-vanishing vector. Substituting representations (3.5.23) and
(3.5.24) into condition (3.5.20) and taking into account that
, (3.5.24)
*'=⋅bUb
, yields condition (3.5.20)
*⋅=UU I
in the form:
'''0
∀⋅⋅>
bbb
n
','0
∈≠
bb
^
Λ . (3.5.25)
But since Λ is the diagonal matrix, condition (3.5.25) is equivalent to
n
where
are eigenvalues, and
λ
k
⎛⎞
λ>
⎜⎟
∑
⎜⎟
1
=
k
⎝⎠
are components of an arbitrary non-vanishing vector 'b.
'
b
k
2
'0
b
kk
, (3.5.26)
The latter condition, as can be easily checked, is equivalent to (3.5.22). Proof for a
matrix A is analogous.
symmetric
Chapter 3. Theory of matrices
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Remark 3.5.7 (Semi-definite matrix)
Sometimes, instead of positive definiteness, the condition of positive
introduced:
Hermitian or symmetric matrix A this is equivalent to
For
Example 3.5.1
A symmetric matrix with only positive components may be neither positive definite, nor
positive semi definite, as the following example shows
Taking vector
and performing the corresponding multiplications, we get
1, 1=−b
()
11 8
()
∀⋅⋅≥
bbAb
n
,0
∈≠
bb
^
Sp( ) 0
−⋅ ⋅ =−
⊂∪A \
15
⎛⎞
=
A
⎜⎟
51
⎝⎠
15 1
⎛⎞⎛⎞
⎜⎟⎜⎟
51 1
⎝⎠⎝⎠
. (3.5.28)
+
. (3.5.29)
−
semi definiteness is
. (3.5.27)
0
. (3.5.30)
Checking eigenvalues of this matrix, we get
Proposition 3.5.9
Hermitian or symmetric matrix (not necessary positive definite or positive semi
If
is a
∈A
n
definite), then its maximum (
λ= ⋅⋅
λ= ⋅⋅
Proof
The proof relies on the procedure used in the proof of Proposition 3.5.8 on choosing an
auxiliary unit vector 'b related to vector
λ= ⋅⋅
max
4, 6λ=− λ =
12
) and minimum (
λ
max
max
min
sup
n
∈=
bb
^
,1
inf
n
∈=
bb
^
,1
by (3.5.24). Then (3.5.32) becomes
b
sup ' '
n
∈=
^
','1
bb
. (3.5.31)
) eigenvalues can be obtained by
λ
min
bAb
()
. (3.5.32)
bAb
()
bb
Λ
()
A
λ= ⋅⋅
min
inf ' '
n
∈=
^
','1
bb
bb
()
Λ
A
(3.5.33)
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