Page 269 - Advanced Linear Algebra

Structure Theory for Normal Operators 253

) ²#³ ) )~ ) # )~ ) #
Thus, if 8 is an orthonormal basis for im ² ³ , then
~¸ Á Ã Á ¹
8 ~¸ Á Ã Á ¹ is an orthonormal basis for ²im ² ³³ ~ im ² ³. Finally, we

may extend both orthonormal bases to orthonormal bases for and then extend
=

the definition of to an isometry on for which ~ .
=
As for the uniqueness, we have seen that must satisfy ~ i and since

has a unique positive square root, we deduce that is uniquely defined. Finally,

if is invertible, then so is since ker²³ ‹ ker²³ . Hence, ~ c is

uniquely determined by .

i
)
Part 2 can be proved by applying the previous theorem to the map , to get

ii
i
~² ³ ~² ³ ~ c ~

where is unitary.

We leave it as an exercise to show that any unitary operator has the form

~ , where is a self-adjoint operator. This gives the following corollary.
Corollary 10.26 (Polar decomposition ) Let be a nonzero linear operator on

a finite-dimensional complex inner product space. Then there is a positive
operator and a self-adjoint operator for which has the polar
decomposition
~

Moreover, is unique and if is invertible, then is also unique.

Normal operators can be characterized using the polar decomposition.
Theorem 10.27 Let ~ be a polar decomposition of a nonzero linear
operator . Then is normal if and only if ~ .

Proof. Since

i
~ c ~

and

i

~ c ~ c
we see that is normal if and only if

c ~