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208 ALGEBRA

5.6.1-9. Decomposition theorems.

THEOREM 1. For any bounded linear operator A in a Hilbert space V, the operator
∗
∗
H 1 = 1 2 (A + A ) is Hermitian and the operator H 2 = 1 2 (A – A ) is skew-Hermitian. The
representation of A as a sum of Hermitian and skew-Hermitian operators is unique: A =
H 1 + H 2 .
THEOREM 2. For any bounded linear operator A in a real Hilbert space, the operator
1
1
T
T
S 1 = (A + A ) is symmetric and the operator S 2 = (A – A ) is skew-symmetric. The
2
2
representation of A as a sum of symmetric and skew-symmetric operators is unique: A =
S 1 + S 2 .
THEOREM 3. For any bounded linear operator A in a Hilbert space, AA and A A are
∗
∗
nonnegative Hermitian operators.
THEOREM 4. For any linear operator A in a Hilbert space V,there exist polar decom-
positions
A = QU and A = U 1 Q 1 ,
2 ∗ 2 ∗
where Q and Q 1 are nonnegative Hermitian operators, Q = AA , Q = A A,and U, U 1
1
are unitary operators. The operators Q and Q 1 are always unique, while the operators U
and U 1 are unique only if A is nondegenerate.
5.6.2. Linear Operators in Matrix Form
5.6.2-1. Matrices associated with linear operators.

Let A be a linear operator in an n-dimensional linear space V with a basis e 1 , ... , e n .Then
j
there is a matrix [a ] such that
j
n

i
Ae j = a e i .
j
i=1
j
The coordinates y of the vector y = Ax in that basis can be represented in the form
n
i j

i
y = a x (i = 1, 2, ... , n), (5.6.2.1)
j
j=1
j
i
where x are the coordinates of x in the same basis e 1 , ... , e n . The matrix A ≡ [a ]ofsize
j
n × n is called the matrix of the linear operator A in a given basis e 1 , ... , e n .
Thus, given a basis e 1 , ... , e n , any linear operator y = Ax can be associated with its
matrix in that basis with the help of (5.6.2.1).
If A is the zero operator, then its matrix is the zero matrix in any basis. If A is the unit
operator, then its matrix is the unit matrix in any basis.
i
THEOREM 1. Let e 1 , ... , e n be a given basis in a linear space V and let A ≡ [a ] be a
j
given square matrix of size n × n. Then there exists a unique linear operator A : V→ V
whose matrix in that basis coincides with the matrix A.
THEOREM 2. The rank of a linear operator A is equal to the rank of its matrix A in any
basis: rank (A)= rank (A).
THEOREM 3. A linear operator A : V→ V is invertible if and only if rank (A)=dim V .
In this case, the matrix of the operator A is invertible.