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Chapter 3. Theory of matrices
121
__________________________________________________________________________
According to (3.2.29) the Frobenius norm is
01
A . (3.2.36)
⎜⎟
F
i
10
⎝⎠
−−
01
⎛⎞
≡⋅⋅−−==
Mind that in (3.2.36) we perform full convolution with respect to two indices. Similarly to
Frobenius norm, computing the generalized Frobenius 1-norm yields
nm
≡ = +++++ =
A
∑∑
F
,1
kl
==
11
a
kl
According to (3.2.32), the generalized Frobenius ∞-norm is
⎛⎞
i
⎜⎟
⎜⎟
⎜⎟
⎝⎠
011110 4
()
i
142
i
0
. (3.2.37)
≡=A
,
∞
F
max 1
a
,
kl
. (3.2.38)
kl
Computation of the nuclear norm (3.2.33) for our matrix yields
⎛⎞
⎜⎟
01 2
−−
≡⋅−−=
A
*
⎛⎞ ⎛⎞
tr 1 tr
⎜⎟
⎜⎟ ⎜⎟
10 2
ii
⎝⎠ ⎝⎠
⎜⎟
⎝⎠
01
⎛⎞
ii
⎜⎟
⎜⎟
⎜⎟
⎝⎠
i
0
i
⎛⎞
⎜⎟
⎜⎟
−
⎝⎠
. (3.2.39)
to proceed further we need to compute the square root of the matrix
22i
⎛⎞
⎜⎟
−
⎝⎠
. (3.2.40)
i
This, according to the methodology described in Sec.3.6.3, needs in reducing matrix (3.2.40)
into its Jordan normal form:
11
2101
⎛⎞ ⎛⎞⎛⎞
⎜⎟ ⎜⎟⎜⎟
ii
−
⎝⎠ ⎝⎠⎝⎠
⎛⎞
ii
2031
22
=⋅⋅
⎜⎟
11
ii
−
22
⎝⎠
−
(3.2.41)
and taking the square root of the diagonal matrix, thus
11
2101
⎛⎞ ⎛⎞
⎜⎟ ⎜⎟
ii
−
⎝⎠ ⎝⎠
⎛⎞
ii
21
22
=⋅⋅=
⎜⎟
11
ii
−
22
⎝⎠
11
⎛⎞
(1 3) ( 1 3)
+−+
22
⎜⎟
⎜⎟
11
(1 3 ) (1 3)
i
−+
22
⎝⎠
⎛⎞
⎜⎟
⎜⎟
03
⎝⎠
−
i
. (3.2.42)
Now, it remains to compute
A
≡=+
*
of the resultant matrix; this yields:
tr
11
⎛⎞
(1 3 ) ( 1 3)
+−+
22
tr 1 3
⎜⎟
⎜⎟
11
(1 3) (1 3)
i
−+
22
⎝⎠
i
. (3.2.43)
It is obvious that for arbitrary matrix the nuclear norm needs in much more computations for its
evaluation.
Chapter 3. Theory of matrices
p
p
p
p
p
122
__________________________________________________________________________
Definition 3.2.8 (Schatten
-norm)
Let
M∈A ^
be arbitrary complex square matrix and
()
n
eigenvalues (see Sec.3.2). The Schatten
Remark 3.2.7
For square matrices Schatten
Definition 3.2.9 (Consistent norms)
Any matrix
A matrix norm
()
M∈A ^ can be considered as linear operator from vector space
,
nm
is called consistent with vector norms
αβ
respectively, if
, 1,...,
knλ=
k
be the corresponding
-norm is defined by:
n
⎛⎞
A . (3.2.44)
≡λ
⎜⎟
∑
,
Sp
⎜⎟
1
k =
⎝⎠
1/
p
k
-norms coincide with Frobenius p-norms.
m
\ into n\ .
⋅≤Ax A x
ααββ
n
and
α
β
on
\
and
. (3.2.45)
\
m
Definition 3.2.10 (Equivalence of norms)
To matrix norms
there exist two nontrivial constants
for any matrix
and
∗
1
()
M∈A ^
,
nm
defined on a matrix space
∗
2
and
k
1
kk≤≤AA A
12
12 1
such that
k
2
(3.2.46)
.
Proposition 3.2.5 (Consistency and equivalence of matrix norms)
Frobenius p-norms at
1 p≤<∞
, nuclear, and Schatten p-norms at
and equivalent.
Remarks 3.2.8
A. Neither of the considered norms are consistent at
It should also be noted that neither of the matrix norms guarantee that the corresponding
B.
=∞
(square) matrix in non-degenerate.
M ^
nm
.
are called equivalent, if
()
,
1 p≤<∞
are consistent
Chapter 3. Theory of matrices
p
p
M
M
123
__________________________________________________________________________
Example 3.2.2
To demonstrate condition of Remark 3.2.8, consider the following matrix
000
⎛⎞
is degenerate, since its determinant obviously vanishes. However, neither of the norms of this
matrix vanishes. We will show that for the Schatten
At first, we construct the characteristic polynomial by its definition (3.2.6) and find its roots
⎜⎟
=
A
000
⎜⎟
⎜⎟
003
⎝⎠
(3.2.47)
-norm.
Now, from (3.2.44) we get
Thus, degenerate matrix (3.2.47) has the non-vanishing Schatten
12 3
p∀=A
0, 3λ=λ = λ=
3.3. Equation Chapter 3 Section 3 Simple
and semisimple matrices
3.3.1. Basic definitions
Definition 3.3.1 (Simple matrix; semisimple matrix)
Matrix
A.
is called simple matrix, if all its eigenvalues are different.
∈A
n
,
Sp
(3.2.48)
. (3.2.49)
3
-norm.
Matrix
B.
eigenvectors.
Proposition 3.3.1
For a simple matrix its characteristic polynomial has no multiple roots.
A.
For a semisimple matrix its minimal polynomial has no multiple roots.
B.
Proof
Proof of Condition A flows out from directly from expression (3.2.6) for the characteristic
polynomial. Proof of condition B is much more complicated and is based on analyzing structure
of the elementary divisors for the minimal polynomial.
is called semisimple, if it has n right (left) linear independent
∈A
n
Chapter 3. Theory of matrices
M
M
124
__________________________________________________________________________
Corollary
If characteristic polynomial has no multiple roots (the corresponding matrix is simple), then the
characteristic polynomial coincides with the minimal polynomial.
Example 3.3.1
Even matrix of a very primitive structure can be not a simple matrix. Let
(diagonal) matrix:
then according to (3.2.6), the characteristic polynomial is
while the minimal polynomial is
So, this matrix is semisimple one due to Proposition 3.3.1. Direct verification shows that the
following 2-dimensional vectors are eigenvectors for this matrix:
Example 3.3.2
The following matrix
be the unit
∈I
2
10
⎛⎞
=
I
⎜⎟
⎝⎠
() det( ) (1 )ψλ ≡ −λ = −λII , (3.3.2)
, (3.3.1)
01
2
() 1ϕλ = −λ. (3.3.3)
10
⎛⎞ ⎛⎞
,
==
mm
12
⎜⎟ ⎜⎟
01
⎝⎠ ⎝⎠
is neither simple, nor semisimple:
∈A
2
. (3.3.4)
21
⎛⎞
=
A
⎜⎟
02
⎝⎠
. (3.3.5)
The characteristic polynomial of this matrix is:
() det( ) (2 )ψλ ≡ −λ = −λAI . (3.3.6)
2
It can easily be verified that for this matrix its minimal polynomial coincides with the
characteristic polynomial. Thus, both these polynomials have multiple roots. According to
Definition 3.3.1 and Proposition 3.3.1, this matrix being not a semisimple matrix, cannot have
two eigenvectors. Direct verification shows that its unique eigenvector is:
1
⎛⎞
m
. (3.3.7)
=
⎜⎟
0
⎝⎠
Chapter 3. Theory of matrices
M
M
R
R
R
R
125
__________________________________________________________________________
Remark 3.3.1
The preceding examples reveal that the simplicity of a matrix is not a very special
A.
property, and it does not affect the number of eigenvectors. At the same time, the
semisimplicity is one of the most important properties of matrices ensuring existence of
the complete set of linearly independent eigenvectors for the given matrix.
According to Proposition 2.2.3, a matrix is semisimple if algebraic multiplicity coincides
B.
with geometric multiplicity for any of its eigenvalues.
3.3.2. Properties of semisimple matrices
Proposition 3.3.2
Let
be a semisimple matrix and
∈A
n
corresponding right and left eigenvectors of matrix
are stored in matrix
raw, then matrices
commutes with matrix
columnwise, while left eigenvectors are stored in matrix
W
L
are defined up to an arbitrary non-degenerate matrix V, which
,
WW
:
A
Proof
By introducing matrix
, identities (3.1.14), and (3.1.15) can be written in the following
W
forms:
and
where
diag( ,..., )
=λλΛ
1
is the diagonal matrix composed of the eigenvalues of matrix A.
n
Now, in view of (3.3.8), identity (3.3.9) implies
⋅⋅ = ⋅⋅ =⋅ ⋅AVW VAW VW Λ
RRR
RL
,
be square matrices composed by the
∈WW
n
A , in which connection right eigenvectors
L
raw by
W
⋅=⋅AV VA. (3.3.8)
RR
⋅=⋅AW W Λ
LL
⋅=⋅WA WΛ
(3.3.9)
, (3.3.10)
. (3.3.11)
Thus, matrix
for matrix
Corollary 1
where as before,
⋅VW
L
is analogous.
W
V is a matrix commuting with
can also be regarded as a matrix composed of the right eigenvectors. Proof
LR
⋅=WW V
, (3.3.12)
.
A
Chapter 3. Theory of matrices
M
M
R
M
R
R
R
126
__________________________________________________________________________
Corollary 2
Since the identity matrix
L
complete sets
W and RW of the eigenvectors can be chosen in such a way that
commutes with any (semisimple) matrix
∈I
n
LR
⋅=WW I. (3.3.13)
Thus, left and right eigenvectors corresponding to different eigenvalues can be made mutually
orthogonal.
Scholium 3.3.1
A. Proposition 3.3.2 shows that both left and right eigenvectors are defined ambiguously. If
L
W and RW are the corresponding eigenmatrices composed of all eigenvectors of a
semisimple matrix
A , and
is a matrix commuting with A , then
V
become also left and right eigenmatrices correspondingly.
B.
In the subsequent analyses, eigenmatrices
W
L
and
W
will be chosen mutually inverse, satisfying condition (3.3.13).
Proposition 3.3.3 (The Jordan normal form for a semisimple matrix)
Any semisimple matrix
can be represented in a form:
∈A
n
, the
∈A
n
L
⋅WV and R⋅VW
for arbitrary semisimple matrix
where
taken according to their multiplicity.
Proof
The proof follows from expression (3.3.9) or (3.3.10).
Remark 3.3.2
The right-hand side of representation (3.3.14) for a semisimple matrix is called the
A.
Matrix
B.
The inverse to Proposition 3.3.3 is also true.
C.
Matrix
D.
1−
=⋅⋅AW WΛ . (3.3.14)
is a non-degenerate matrix, and
W
=diag( ,..., )λλΛ
1n
with eigenvalues
Jordan normal (or canonical) form.
in (3.3.14) is defined up to an arbitrary matrix V commuting with matrix A.
W
can be chosen as
W
W
L
and
W
1−
as
W
, where
W
L
and
condition (3.3.13). In such a case representation (3.3.14) takes the form:
L
=⋅⋅AW WΛ
. (3.3.15)
of matrix A
λ
k
satisfy
W
Chapter 3. Theory of matrices
M
M
M
127
__________________________________________________________________________
Example 3.3.3
Herein we consider the complete set of all right eigenvectors for the following matrix
12
⎛⎞
=
A
⎜⎟
34
⎝⎠
. (3.3.16)
:
∈A
2
Firstly, its characteristic polynomial is:
−λ
12
ψ λ = = λ− λ−
() det ( )( )
⎛⎞
⎜⎟
34
⎝⎠
−λ
+−
533 533
. (3.3.17)
22
The minimal polynomial for this matrix coincides with the characteristic polynomial, and since
the elementary divisors are linear, this matrix is
simple. Now, the complete set of its right-
eigenvectors is:
⎛⎞
333 333
−+ −−
R
W
⎜⎟
=
66
⎜⎟
⎜⎟
11
⎝⎠
, (3.3.18)
while the set of its left eigenvectors is:
⎛⎞
33 11 33
⎜⎟
L
W . (3.3.19)
11 22
⎜⎟
=
⎜⎟
33 11 33
−
⎜⎟
⎝⎠
11 22
+
−
Both these sets are chosen to satisfy the inverse condition (3.3.13). Now, according to (3.3.15),
matrix
has the Jordan normal form (3.3.15):
A
⎛⎞
⎜⎟
⎜⎟
⎜⎟
⎝⎠
Proposition 3.3.4
A sufficient condition for two
same set of all left (or right) eigenvectors.
Proof
Let
,
∈AB
(3.3.15) their Jordan normal forms are:
Now, taking into account (3.3.21), we get
⎛⎞⎛⎞
533 331133
−+ −−
333 333
66
11
⎜⎟⎜⎟
21122
⎜⎟⎜⎟
⋅⋅
⎜⎟⎜⎟
0
⎜⎟⎜⎟
⎝⎠⎝⎠
semisimple matrices
have the same set of left (or right) eigenvectors, then in view of (3.3.14),
n
11
++
AW W BW W
11
⋅= ⋅⋅⋅ ⋅= ⋅⋅⋅
ABWWBAWW
−−
ΛΛ
ΛΛ ΛΛ
ΑΑ
−−
=⋅⋅ =⋅⋅
,
Α
,
BB
0
. (3.3.20)
533 331133
−−
−
21122
,
to commute, is existence of the
∈AB
n
. (3.3.21)
B
. (3.3.22)
Chapter 3. Theory of matrices
M
M
128
__________________________________________________________________________
But, all the diagonal matrices commute (it can easily be verified), so ⋅=⋅
Remark 3.3.3
The inverse condition to the preceding proposition is not true: two commuting semisimple
matrices can have different sets of the corresponding eigenvectors.
Proposition 3.3.5
AB BA.
Proof
If
A.
is an invertible semisimple matrix then
∈A
n
Both A and
matrix
W (in view of Corollary 2 to Proposition 3.3.2 this means that both A and
1−
can be reduced to the Jordan normal form by the same transform
A
A
have the same sets of eigenvectors).
The corresponding diagonal matrices
B.
Λ=Λ
11−
A
()
and 1−ΛA satisfy a relation:
Λ
A
−
(3.3.23)
A
Both of the conditions can be proved by the direct constructing the inverse matrix in the form:
11
−−
=⋅⋅
AW WΛ . (3.3.24)
1
−
A
3.4. Equation Chapter 3 Section 4
Non-semisimple matrices
1−
Definition 3.4.1 (Jordan block)
A square matrix
∈J
nn
3.4.1. Basic definitions
is called the Jordan block of the rank n, if this matrix is of the form:
Chapter 3. Theory of matrices
M
M
p
p
129
__________________________________________________________________________
1
λ
⎛⎞
⎜⎟
⎜⎟
⎜⎟
=
J . (3.4.1)
n
⎜⎟
⎜⎟
⎜⎟
⎝⎠
Definition 3.4.2 (Nilpotent matrix; Index of nilpotency)
1
λ
...
1
λ
λ
A square matrix
index of nilpotency), if
the
∈N
n
is called the
Example 3.4.1
The following examples deliver
nilpotent matrices of the second and third index of nilpotency,
respectively:
01
⎛⎞
==
NN
23
⎜⎟
00
⎝⎠
Definition 3.4.3 (Generalized eigenvector)
Let
∈A
, vector
n
is called the generalized (right) eigenvector of the index p, if
m
ker and ker
∈−λ ∉−λmAI mAI
pp
() ()
nilpotent matrix of the index
0, 0
≠=NN. (3.4.2)
k
010
⎛⎞
,001
⎜⎟
⎜⎟
⎜⎟
000
⎝⎠
. (3.4.3)
(sometimes k is called as
k
1
p
−
. (3.4.4)
Remark 3.4.1
A. The preceding definition along with Definition 2.1.6 mean:
It follows from (3.4.5) that at
B.
Example 3.4.2
The following example shows that a non-semisimple matrix can have several linearly
independent generalized eigenvectors. Moreover, the space of the generalized eigenvectors of
the same index can form a basis.
Let
pp
−λ ⋅ = −λ ⋅ ≠AIm AI m
() ()
0, while 0
pp
genuine (right) eigenvector.
1
−
the generalized eigenvector coincides with the
1p =
. (3.4.5)
Chapter 3. Theory of matrices
M
130
__________________________________________________________________________
110
⎛⎞
then the corresponding characteristic polynomial is
() det( ) (1 )ψλ ≡ −λ = −λAI
⎜⎟
=
A
011
⎜⎟
⎜⎟
001
⎝⎠
, (3.4.6)
3
, (3.4.7)
with three (multiple) eigenvalues
−λAI becomes
k
AI
1, 1, 2 , 3
k
−λ =
k
kλ= =
010
⎛⎞
⎜⎟
001
⎜⎟
⎜⎟
000
⎝⎠
. At any of these eigenvalues the matrix
. (3.4.8)
Direct verification shows that vector
1
⎛⎞
⎜⎟
m
(3.4.9)
=
0
⎜⎟
⎜⎟
0
⎝⎠
is the genuine (right) eigenvector, since it satisfies (3.2.4). At the same time, vectors
00
⎛⎞ ⎛⎞
⎜⎟ ⎜⎟
are not eigenvectors, since they do not satisfy (3.2.4). But, taking in (3.4.4)
−λ = ⋅ =
AI
()
k
′′′
==
mm
10
⎜⎟ ⎜⎟
⎜⎟ ⎜⎟
01
⎝⎠ ⎝⎠
010 010 001
⎛⎞⎛⎞⎛⎞
2
⎜⎟⎜⎟⎜⎟
001 001 000
⎜⎟⎜⎟⎜⎟
⎜⎟⎜⎟⎜⎟
000 000 000
⎝⎠⎝⎠⎝⎠
(3.4.10)
2p = , yields
. (3.4.11)
The right-hand side of (3.4.11) reveals that both
matrix
()
eigenvectors of the index 2.
Proposition 3.4.1
A.
Let
2
−λAI
. Thus, according to Definition 3.4.3,
k
∈J
nn
′
and
m
m
m
′
and
3.4.2. Basic properties
be the Jordan block of the n -th rank, then
=λ +JIN
nn
, (3.4.12)
′′
belong to the kernel space of
′′
m
become the generalized
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