Page 316 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes If f is a complex-valued function whose domain includes c , ..., c , then
1 k
f(A) = f(c )E + ... + f(c )E ;
1 1 k k
in particular, A = c E + ... + c E .
1 1 k k
We recall that an operator on an inner product space V is non-negative if T is self-adjoint and
(T | ) 0 for every in V.
Theorem 4: Let T be a diagonalizable normal operator on a finite-dimensional inner product
space V. Then T is self-adjoint, non-negative, or unitary according as each characteristic value of
T is real, non-negative, or of absolute value 1.
Proof: Suppose T has the spectral resolution T = c E + ... + c E , then T* = c E + ... + c E . To say
1 1 k k 1 1 k k
T is self-adjoint is to say T = T*, or
(c – c )E + ... + (c – c )E = 0.
1 1 1 k k k
Using the fact that E E = 0 for i j, and the fact that no E, is the zero operator, we see that T is
i j j
self-adjoint if and only if c = c , j = 1, ... , k. To distinguish the normal operators which are
j j
non-negative, let us look at
k k
(T | ) = c E | E i
j j
j 1 i 1

= c j (E j |E i )
i j
2
c E
= j j
j
We have used the fact that (E |E ) = 0 for i j. From this it is clear that the condition (T | ) 0
j i
is satisfied if and only if c 0 for each j. To distinguish the unitary operators, observe that
j
TT* = c c E + ... + c c E .
1 1 1 k k k
= c E + ... + c E .
2
2
1 1 k k
2
2
If TT* = I, then I = |c | E + ... + c E , and operating with E
1 1 k k j
2
Ej = c E .
j j
2
Since E 0, we have c = 1 or c = l. Conversely, if c = 1 for each j it is clear that TT* = I.
2
j j j j
It is important to note that this is a theorem about normal operators. If T is a general linear
operator on V which has real characteristic values, it does not follow that T is self-adjoint. The
theorem states that if T has real characteristic values, and if T is diagonalizable and normal, then
T is self-adjoint. A theorem of this type serves to strengthen the analogy between the adjoint
operation and the process of forming the conjugate of a complex number. A complex number z
is real or of absolute value 1 according as z = z , or z z = 1. An operator T is self-adjoint or unitary
according as T = T* or T*T = I.
We are going to prove two theorems now, which are the analogues of these two statements:
1. Every non-negative number has a unique non-negative square root.
2. Every complex number is expressible in the form ru, where r is non-negative and u = 1.
This is the polar decomposition z = re for complex numbers.
i

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