Page 259 - Advanced Linear Algebra

Structure Theory for Normal Operators 243

The analog of Theorem 2.19 is the following.

Theorem 10.15 Let be an inner product space of dimension . Then two
=

d matrices ( and ) are unitarily/orthogonally similar if and only if they
(
B
represent the same linear operator ²= ³ with respect to possibly different)

ordered orthonormal bases. In this case, and represent exactly the same
(
)
set of linear operators in B²= ³ with respect to ordered orthonormal bases.
)
(
Proof. If and represent  ² = ³B , that is, if
and
( ~ ´µ 8 ) ~ ´µ 9
for ordered orthonormal bases and , then
9
8
)~ 4 89Á (4 9 8Á
and according to Theorem 10.14, 4 is unitary/orthogonal. Hence, and )
( 89 Á
are unitarily/orthogonally similar.
Now suppose that and are unitarily/orthogonally similar, say
(
)
)~ <(< c
where is unitary/orthogonal. Suppose also that represents a linear operator
(
<
B ²= ³ for some ordered orthonormal basis , that is,
8

(~ ´ µ 8
Theorem 10.14 implies that there is a unique ordered orthonormal basis for =
9
. Hence
for which <~ 4 89Á
c

8
)~ 4 89Á ´ µ 4 89 ~ ´ µ 9
Á
and so also represents . By symmetry, we see that and represent the
(

)
)
same set of linear operators, under all possible ordered orthonormal bases.
We have shown (see the discussion of Schur's theorem) that any complex matrix
( ( is unitarily similar to an upper triangular matrix, that is, that is unitarily
upper triangularizable. However, upper triangular matrices do not form a set of
canonical forms under unitary similarity. Indeed, the subject of canonical forms
for unitary similarity is rather complicated and we will not discuss it in this
book, but instead refer the reader to the survey article [28].
Reflections
The following defines a very special type of unitary operator.