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170 ALGEBRA

Properties of multiplication of matrices:
AO = O 1 , A + O = A (property of zero matrix),
(AB)C = A(BC) (associativity of the product of three matrices),
AI = A (multiplication by unit matrix),
A(B + C)= AB + AC (distributivity with respect to a sum of two matrices),
λ(AB)= (λA)B = A(λB) (associativity of the product of a scalar and two matrices),
SD = DS (commutativity for any square and any diagonal matrices),
where λ is a scalar, matrices A, B, C, square matrix S, diagonal matrix D, zero matrices O
and O 1 , and unit matrix I have the compatible sizes.

5.2.1-3. Transpose, complex conjugate matrix, adjoint matrix.
The transpose of a matrix A ≡ [a ij ]ofsize m × n is the matrix C ≡ [c ij ]of size n × m with
entries
c ij = a ji (i = 1, 2, ... , n; j = 1, 2, ... , m).
T
The transpose is denoted by C = A .
a 1
T
Example 2. If A =(a 1, a 2)then A = .
a 2
Properties of transposes:
T
T T
T
T
T
T
(A + B) = A + B , (λA) = λA , (A ) = A,
T
T
T
T
T
(AC) = C A , O = O 1 , I = I,
where λ is a scalar; matrices A, B, and zero matrix O have size m × n;matrix C has size
n × l; zero matrix O 1 has size n × m.
T
T
T
A square matrix A is said to be orthogonal if A A = AA = I, i.e., A = A –1 (see
Paragraph 5.2.1-6).
Properties of orthogonal matrices:
T
1. If A is an orthogonal matrix, then A is also orthogonal.
2. The product of two orthogonal matrices is an orthogonal matrix.
3. Any symmetric orthogonal matrix is involutive (see Paragraph 5.2.1-7).
The complex conjugate of a matrix A ≡ [a ij ]of size m × n is the matrix C ≡ [c ij ]of
size m × n with entries
c ij = ¯a ij (i = 1, 2, ... , m; j = 1, 2, ... , n),
where ¯a ij is the complex conjugate of a ij . The complex conjugate matrix is denoted
by C = A.
The adjoint matrix of a matrix A ≡ [a ij ]ofsize m × n is the matrix C ≡ [c ij ]of size
n × m with entries
c ij = ¯a ji (i = 1, 2, ... , n; j = 1, 2, ... , m).
The adjoint matrix is denoted by C = A .
∗
Properties of adjoint matrices:
¯
∗ ∗
∗
∗
∗
∗
(A + B) = A + B ,(λA) = λA , (A ) = A,
∗
∗
∗
(AC) = C A , O = O 1 , I = I,
∗
∗
∗
where λ is a scalar; matrices A, B, and zero matrix O have size m × n;matrix C has size
n × l; zero matrix O 1 has a size n × m.