5.2. MATRICES AND DETERMINANTS 171
Remark. If a matrix is real (i.e., all its entries are real), then the corresponding transpose and the adjoint
matrix coincide.
A square matrix A is said to be normal if A
∗
A = AA
∗
. A normal matrix A is said to be
unitary if A
∗
A = AA
∗
= I, i.e., A
∗
= A
–1
(see Paragraph 5.2.1-6).
5.2.1-4. Trace of a matrix.
The trace of a square matrix A ≡ [a
ij
]ofsizen × n is the sum S of its diagonal entries,
S =Tr(A)=
n

i=1
a
ii
.
If λ is a scalar and square matrices A and B has the same size, then
Tr(A + B)=Tr(A)+Tr(B), Tr(λA)=λTr(A), Tr(AB)=Tr(BA),
5.2.1-5. Linear dependence of row vectors (column vectors).

1
+ ···+ α
2
k
≠ 0) such that
α
1
A
1
+ ···+ α
k
A
k
= O,
where O is the zero row vector (column vector).
Row vectors (column vectors) A
1
, , A
k
are said to be linearly independent if, for
any α
1
, , α
k
(α
2
1
+ ···+ α
2
k

A square matrix is nondegenerate if and only if its rows (columns) are
linearly independent.
Remark. Generally, instead of the terms “left inverse matrix” and “right inverse matrix”, the term “inverse
matrix” is used with regard to the matrix B = A
–1
for a nondegenerate matrix A,sinceAB = BA = I.
UNIQUENESS THEOREM.
The matrix
A
–1
is the unique matrix satisfying the condition
AA
–1
= A
–1
A = I
for a given nondegenerate matrix
A
.
Remark. For the existence theorem, see Paragraph 5.2.2-7.
172 ALGEBRA
Properties of inverse matrices:
(AB)
–1
= B
–1
A
–1
,(λA)
–1

A product of several matrices equal to one and the same matrix A can be written as a positive
integer power of the matrix A: AA = A
2
, AAA = A
2
A = A
3
, etc. For a positive integer k,
one defines A
k
= A
k–1
A as the kth power of A. For a nondegenerate matrix A, one defines
A
0
= AA
–1
= I, A
–k
=(A
–1
)
k
. Powers of a matrix have the following properties:
A
p
A
q
= A
p+q

2
+ ···,
where a
i
(i = 0, 1, 2, ) are real or complex coefficients. The polynomial f(X) is a square
matrix of the same size as X.
A polynomial with matrix coefficients is an expression obtained from a polynomial f (x)
by replacing its coefficients a
i
(i = 0, 1, 2, ) with matrices A
i
(i = 0, 1, 2, )ofthe
same size:
F (x)=A
0
+ A
1
x + A
2
x
2
+ ···.
Example 3. For the matrix
A =

4 –81
5 –91
4 –6 –1

,

The corresponding adjugate matrix (see Paragraph 5.2.2-7) can also be represented as a polynomial with matrix
coefficients:
G(λ)=

λ
2
+ 10λ + 15 –8λ – 14 λ + 1
5λ + 9 λ
2
– 3λ – 8 λ + 1
4λ + 6 –6λ – 8 λ
2
+ 5λ + 4

= A
0
+ A
1
λ + A
2
λ
2
,
5.2. MATRICES AND DETERMINANTS 173
where
A
0
=

15 –14 1

2
+ ···,

F (X)=A
0
+ XA
1
+ X
2
A
2
+ ···.
The exponential function of a square matrix X can be represented as the following
convergent series:
e
X
= 1 + X +
X
2
2!
+
X
3
3!
+ ···=
∞

k=0
X
k

Y
≠ e
Y
e
X
, in general. The relation e
X
e
Y
= e
X+Y
holds only for commuting
matrices X and Y .
Some other functions of matrices can be expressed in terms of the exponential function:
sin X =
1
2i
(e
iX
– e
–iX
), cos X =
1
2
(e
iX
+ e
–iX
),
sinh X =

2
=
1
2
(A – A
T
)
is skew-symmetric. The representation of
A
as the sum of
symmetric and skew-symmetric matrices is unique:
A = S
1
+ S
2
.
THEOREM 2.
For any square matrix
A
, the matrices
H
1
=
1
2
(A+A
∗
)
and
H

Hermitian matrices (see Paragraph 5.7.3-1).
THEOREM 4.
Any square matrix
A
admits a
polar decomposition
A = QU
and
A = U
1
Q
1
,
where
Q
and
Q
1
are nonnegative Hermitian matrices,
Q
2
= AA
∗
and
Q
2
1
= A
∗
A

, i
α
+ 1, , i
α
+ m
α
– 1; j = j
β
, j
β
+ 1, , j
β
+ n
β
– 1)of
size m
α
× n
β
and is called a block of the matrix A.Herei
α
= m
α–1
+ i
α–1
, j
β
= n
β–1
+ j

24
a
25
a
31
a
32
a
33
a
34
a
35
a
41
a
42
a
43
a
44
a
45
a
51
a
52
a
53
a

13
a
21
a
22
a
23

, A
12
≡

a
14
a
15
a
24
a
25

,
A
21
≡

a
31
a
32

55

of size 2×3, 2×2, 3×3, 3×2, respectively.
Basic operations with block matrices are practically the same as those with common
matrices, the role of the entries being played by blocks:
1. For matrices A ≡ [a
ij
] ≡ [A
αβ
]andB ≡ [b
ij
] ≡ [B
αβ
] of the same size and the same
block structure, their sum C ≡ [C
αβ
]=[A
αβ
+ B
αβ
] is a matrix of the same size and
the same block structure.
2. For a matrix A ≡ [a
ij
]ofsizem× n regarded as a block matrix A ≡ [A
αβ
]ofsizeM × N,
the multiplication by a scalar is defined by λA =[λA
αβ
]=[λa

αβ
]ofsizeM × N,
the transpose has the form A
T
=[A
T
βα
].
5. For a matrix A ≡ [a
ij
]ofsizem× n regarded as a block matrix A ≡ [A
αβ
]ofsizeM × N,
the adjoint matrix has the form A
∗
=[A
∗
βα
].
Let A be a nondegenerate matrix of size n × n represented as the block matrix
A ≡

A
11
A
12
A
21
A
22

N
–NA
21
A
–1
11
N

,
A
–1
=

K –KA
12
A
–1
22
–A
–1
22
A
21
KA
–1
22
+ A
–1
22
A

;inthefirst formula, the matrix
A
11
is assumed nondegenerate, and in the second formula, A
22
is assumed nondegenerate.
5.2. MATRICES AND DETERMINANTS 175
The direct sum of two square matrices A and B of size m × m and n × n, respectively,
is the block matrix C = A ⊕ B =

A 0
0 B

of size m + n.
Properties of the direct sum of matrices:
1. For any square matrices A, B,andC the following relations hold:
(A ⊕ B) ⊕ C = A ⊕ (B ⊕ C) (associativity),
Tr(A ⊕ B)=Tr(A)+Tr(B) (trace property).
2. For nondegenerate square matrices A and B, the following relation holds:
(A ⊕ B)
–1
= A
–1
⊕ B
–1
.
3. For square matrices A
m
, B
m

n
)=A
m
B
m
⊕ A
n
B
n
.
5.2.1-11. Kronecker product of matrices.
The Kronecker product of two matrices A ≡ [a
i
a
j
a
]andB ≡ [b
i
b
j
b
]ofsizem
a
× n
a
and
m
b
× n
b

b
),
where the index k is the serial number of a pair (i
a
, i
b
) in the sequence (1, 1), (1, 2), ,
(1, m
b
), (2, 1), (2, 2), (m
a
, m
b
), and the index h is the serial number of a pair (j
a
, j
b
)
in a similar sequence. This Kronecker product can be represented as the block matrix
C ≡ [a
i
a
j
a
B].
Note that if A and B are square matrices and the number of rows in C is equal to the
number of rows in A, and the number of rows in D is equal to the number of rows in B,then
(A ⊗ B)(C ⊗ D)=AC ⊗ BD.
The following relations hold:
(A ⊗ B)

a
22



= a
11
a
22
– a
12
a
21
.
176 ALGEBRA
For a matrix of size 3×3(n = 3), the third-order determinant,
Δ ≡ det A ≡





a
11
a
12
a
13
a
21

32
a
13
– a
13
a
22
a
31
– a
12
a
21
a
33
– a
23
a
32
a
11
.
This expression is obtained by the triangle rule (Sarrus scheme), illustrated by the following
diagrams, where entries occurring in the same product with a given sign are joined by
segments:
+





✁
✁
✁
✁







–















❍
❍
❍

Consider a matrix A =[a
ij
]ofsizen × n.Theminor M
i
j
corresponding to an entry a
ij
is defined as the (n – 1)st-order determinant of the matrix of size (n – 1) × (n – 1) obtained
from the original matrix A by removing the ith row and the jth column (i.e., the row and
the column whose intersection contains the entry a
ij
). The cofactor A
i
j
of the entry a
ij
is
defined by A
i
j
=(–1)
i+j
M
i
j
(i.e., it coincides with the corresponding minor if i + j is even,
and is the opposite of the minor if i + j is odd).
The nth-order determinant of the matrix A is defined by
Δ ≡ det A ≡


.
a
n1
a
n2
··· a
nn








=
n

k=1
a
ik
A
i
k
=
n

k=1
a
kj



65
2 –4



+(–1)
2+2
×1×



12
2 –4



+(–1)
3+2
× (–1) ×



12
65



= 1×[6×(–4)–5×2]+1×[1×(–4)–2×2]+1×[1×5– 2×6]=–49.

c
,
where A
b
and A
c
are the matrices obtained from A by replacing its ith row with
(b
1
, , b
n
)and(c
1
, , c
n
). This fact, together with the first property, implies that a
similar linearity relation holds if a column of the matrix A is a linear combination of
two column vectors.
Some useful corollaries from the basic properties:
1. The determinant of a matrix with two equal rows (columns) is equal to zero.
2. If all entries of a row are multiplied by λ, the determinant of the resulting matrix is
multiplied by λ.
3. If a matrix contains a row (columns) consisting of zeroes, then its determinant is equal
to zero.
4. If a matrix has two proportional rows (columns), its determinant is equal to zero.
5. If a matrix has a row (column) that is a linear combination of its other rows (columns),
its determinant is equal to zero.
6. The determinant of a matrix does not change if a linear combination of some of its rows
is added to another row of that matrix.
T

1
, j
2
, , j
m
are the indices of the columns involved in that
submatrix. The (n – m)th-order determinant of the second kind, denoted by
M
i
1
i
2
i
m
j
1
j
2
j
m
,is
the (n – m)th-order determinant of the submatrix obtained from A by removing the rows
and the columns involved in M
i
1
i
2
i
m
j

j
m
=(–1)
i
1
+i
2
+···+i
m
+j
1
+j
2
+···+j
m
M
i
1
i
2
i
m
j
1
j
2
j
m
.
Remark. minors of the first kind can be introduced for any rectangular matrix A ≡ [a

, , j
m
)ofasquarematrix
A
, its determinant
Δ
is equal to the sum of products of all
m
th-order minors
M
i
1
i
2
i
m
j
1
j
2
j
m
in those rows (resp., columns) and their cofactors
A
i
1
i
2
i
m

1
i
2
i
m
j
1
j
2
j
m
=

i
1
,i
2
, ,i
m
M
i
1
i
2
i
m
j
1
j
2

positive integer r ≤ n for which the following conditions hold:
i) the matrix A has an rth-order nonzero minor, and
ii) any minor of A of order (r + 1) and higher (of it exists) is equal to zero.