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5.2. MATRICES AND DETERMINANTS 175

The direct sum of two square matrices A and B of size m × m and n × n, respectively,
A
0
is the block matrix C = A ⊕ B = of size m + n.
0 B
Properties of the direct sum of matrices:
1. For any square matrices A, B,and C 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:
–1 –1 –1
(A ⊕ B) = A ⊕ B .
3. For square matrices A m , B m of size m × m and square matrices A n , B n of size n × n,
the following relations hold:
(A m ⊕ A n )+ (B m ⊕ B n )= (A m + B m ) ⊕ (A n + B n );
(A m ⊕ A n )(B m ⊕ B n )= A m B m ⊕ A n B n .

5.2.1-11. Kronecker product of matrices.

]of size m a × n a and
The Kronecker product of two matrices A ≡ [a i aj a ]and B ≡ [b i b j b
m b × n b , respectively, is the matrix C ≡ [c kh ]ofsize m a m b × n a n b with entries
b (k = 1, 2, ... , m a m b ; h = 1, 2, ... , n a n b ),
c kh = a i aj a i b j 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
B].
C ≡ [a i aj a
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:
T
T
T
(A ⊗ B) = A ⊗ B , Tr(A ⊗ B)=Tr(A)Tr(B).
5.2.2. Determinants
5.2.2-1. Notion of determinant.

With each square matrix A ≡ [a ij ]of size n×n one can associate a numerical characteristic,
called its determinant. The determinant of such a matrix can be defined by induction with
respect to the size n.
For a matrix of size 1×1 (n = 1), the first-order determinant is equal to its only entry,
Δ ≡ det A = a 11 .
For a matrix of size 2× 2 (n = 2), the second-order determinant, is equal to the
difference of the product of its entries on the main diagonal and the product of its entries
on the secondary diagonal:

a 11
= a 11 a 22 – a 12 a 21 .
a 12
a 21 a 22
Δ ≡ det A ≡