Page 318 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes Let us repeat the definition of U. Since V = W W , each in V is uniquely expressible in the
form = N + , where N is in the range W of N, and is in W . We define
U = T + U .
0
This U is clearly linear, and we verified above that it is well-defined. Also
(U U ) (T + U T + U )
0 0
= (T T ) + (U U )
0 0
= (N N ) + ( )

= ( )
and so U is unitary. We also have UN = T for each .
We call T = UN a polar decomposition for T. We certainly cannot call it the polar decomposition,
since U is not unique. Even when T is invertible, so that U is unique, we have the difficulty that
U and N may not commute. Indeed, they commute if and only if T is normal. For example, if
T = UN = NU, with N non-negative and U unitary, then
2
TT* = (NU)(NU)* = NUU*N = N = T*T.
The general operator T will also have a decomposition T = N U , with N non-negative and U
1 1 1 1
unitary. Here, N will be the non-negative square root of TT*. We can obtain this result by
1
applying the theorem just proved to the operator T*, and then taking adjoints.
We turn now to the problem of what can be said about the simultaneous diagonalization of
commuting families of normal operators. For this purpose the following terminology is
appropriate.

Definition: Let  be a family of operators on an inner product space V. A function r on  with
values in the field  of scalars will be called a root of  if there is a non-zero in V such that
T = r(T)

for all T in . For any function r from  to , let V(r) be the set of all in V such that T = r(T)
for every T in .
Then V(r) is a subspace of V, and r is a root of  if and only if V(r) {0}. Each non-zero in V(r)
is simultaneously a characteristic vector for every T in .
Theorem 7: Let  be a commuting family of diagonalizable normal operators on a finite-
dimensional inner product space V. Then  has only a finite number of roots. If r , ... , r are the
1 k
distinct roots of , then
(i) V(r ) is orthogonal to V(r ) when i j, and
i j
(ii) V = V(r ) ... V(r ).
1 k
Proof: Suppose r and s are distinct roots of . Then there is an operator T in  such that r(T) s(T).
Since characteristic vectors belonging to distinct characteristic values of T are necessarily
orthogonal, it follows that V(r) is orthogonal to V(s). Because V is finite-dimensional, this
implies  has at most a finite number of roots. Let r ,..., r , be the roots of F. Suppose {T , ..., T }
1 k 1 m
is a maximal linearly independent subset of , and let
{E , E , ... }
i1 i2
be the resolution of the identity defined by T , (1 i m). Then the projections E form a
i ij
commutative family. For each E is a polynomial in T and T , ... , T , commute with one another.
ij i 1 m

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