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172 ALGEBRA

Properties of inverse matrices:
–1 –1 –1 –1 –1 –1
(AB) = B A , (λA) = λ A ,
∗ –1
T –1
–1 –1
–1 T
–1 ∗
(A ) = A, (A ) =(A ) , (A ) =(A ) ,
where square matrices A and B are assumed to be nondegenerate and scalar λ ≠ 0.
The problem of finding the inverse matrix is considered in Paragraphs 5.2.2-7, 5.2.4-5,
and 5.5.2-3.
5.2.1-7. Powers of matrices.

A product of several matrices equal to one and the same matrix A can be written as a positive
3
2
2
integer power of the matrix A: AA = A , AAA = A A = A , etc. For a positive integer k,
k
one defines A = A k–1 A as the kth power of A. For a nondegenerate matrix A, one defines
0
–1 k
A = AA –1 = I, A –k =(A ) . Powers of a matrix have the following properties:
pq
p q
p
q
A A = A p+q , (A ) = A ,
where p and q are arbitrary positive integers and A is an arbitrary square matrix; or p and q
are arbitrary integers and A is an arbitrary nondegenerate matrix.
k
There exist matrices A whose positive integer power is equal to the zero matrix, even
k
if A ≠ O.If A = O for some integer k > 1,then A is called a nilpotent matrix.
2
Amatrix A is said to be involutive if it coincides with its inverse: A = A –1 or A = I.
5.2.1-8. Polynomials and matrices. Basic functions of matrices.
A polynomial with matrix argument is the expression obtained from a scalar polynomial f(x)
by replacing the scalar argument x with a square matrix X:
2
f(X)= a 0 I + a 1 X + a 2 X + ··· ,
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:
2
F(x)= A 0 + A 1 x + A 2 x + ··· .
Example 3. For the matrix
4 –8 1
( )
A = 5 –9 1 ,
4 –6 –1
the characteristic matrix (see Paragraph 5.2.3-2) is a polynomial with matrix coefficients and argument λ:
4 – λ
( –8 1 )
F(λ) ≡ A – λI = A 0 + A 1λ = 5 –9 – λ 1 ,
4 –6 –1 – λ
where
4 –8 1 –1 0 0
( ) ( )
A 0 = A = 5 –9 1 , A 1 =–I = 0 –1 0 .
4 –6 –1 0 0 –1
The corresponding adjugate matrix (see Paragraph 5.2.2-7) can also be represented as a polynomial with matrix
coefficients:
2
( λ + 10λ + 15 –8λ – 14 λ + 1 )
2
2
G(λ)= 5λ + 9 λ – 3λ – 8 λ + 1 = A 0 + A 1λ + A 2λ ,
2
4λ + 6 –6λ – 8 λ + 5λ + 4