Page 206 -

5.2. MATRICES AND DETERMINANTS 173

where
15 –14 1 10 –81 100
( ) ( ) ( )
A 0 = 9 –8 1 , A 1 = 5 –31 , A 2 = I = 010 .
6 –8 4 4 –65 001
The variable x in a polynomial with matrix coefficients can be replaced by a matrix X,
which yields a polynomial of matrix argument with matrix coefficients. In this situation,
one distinguishes between the “left” and the “right” values:
2
F(X)= A 0 + A 1 X + A 2 X + ··· ,
2
F(X)= A 0 + XA 1 + X A 2 + ··· .
$

The exponential function of a square matrix X can be represented as the following
convergent series:
2 3 ∞ k
X X X
X
e = 1 + X + + + ··· = .
2! 3! k!
k=0
The inverse matrix has the form
2 X 3 X k
∞
X
X –1
(e ) = e –X = 1 – X + – + ··· = (–1) k .
2! 3! k!
k=0
X Y
Y
X
X Y
Remark. Note that e e ≠ e e , in general. The relation e e = 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:
1 1
sin X = (e iX – e –iX ), cos X = (e iX + e –iX ),
2i 2
1 1
X
X
sinh X = (e – e –X ), cosh X = (e + e –X ).
2 2
5.2.1-9. Decomposition of matrices.
1 T
THEOREM 1. For any square matrix A, the matrix S 1 = 2 (A + A ) is symmetric and
T
the matrix S 2 = 1 2 (A – A ) is skew-symmetric. The representation of A as the sum of
symmetric and skew-symmetric matrices is unique: A = S 1 + S 2 .
1
1
THEOREM 2. For any square matrix A, the matrices H 1 = (A+A ) and H 2 = 2i (A–A )
∗
∗
2
are Hermitian, and the matrix iH 2 is skew-Hermitian. The representation of A as the sum
of Hermitian and skew-Hermitian matrices is unique: A = H 1 + iH 2 .
THEOREM 3. For any square matrix A, the matrices AA and A A are nonnegative
∗
∗
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 ,
2 2
∗
where Q and Q 1 are nonnegative Hermitian matrices, Q = AA and Q = A A,and U
∗
1
and U 1 are unitary matrices. The matrices Q and Q 1 are always unique, while the matrices U
and U 1 are unique only in the case of a nondegenerate A.