Page 263 - Advanced Linear Algebra

Structure Theory for Normal Operators 247

Theorem 10.19 (The structure theorem for normal matrices )
1)(Complex case )
)
a A complex matrix ( is normal if and only if it is unitarily
diagonalizable, that is, if and only if there is a unitary matrix for
<
which
i
<(< ~ diag ² Á Ã Á ³
b A complex matrix is Hermitian if and only if 1a holds, where all
)
)
(
eigenvalues are real.

)
)
c A complex matrix is unitary if and only if 1a holds, where all
(
eigenvalues have norm .

2)(Real case )
a A real matrix is normal if and only if there is an orthogonal matrix
)
(
6 for which

! c c
6(6 ~ diag 6 ÁÃÁ Á > ? ÁÃÁ > ? 7

b A real matrix ( is symmetric if and only if it is orthogonally
)
diagonalizable, that is, if and only if there is an orthogonal matrix 6
for which
!
6(6 ~ diag ² Á Ã Á ³
)
c A real matrix is orthogonal if and only if there is an orthogonal
(
matrix for which
6
6(6 !
sin c sin ccos cos
~ diag 6 Á ÃÁ Á > cos sin ? Á ÃÁ > cos sin ? 7
for some  . s ÁÃÁ
Functional Calculus
Let be a normal operator on a finite-dimensional inner product space and let

=
have spectral resolution
b
~ b Ä
Since each ~ for all ‚ . The pairwise
is idempotent, we have

orthogonality of the projections implies that
~² b Ä b ³ ~ b Ä b

More generally, for any polynomial ²%³ over ,
-

² ³ ~ ² ³ b Ä b ² ³
Note that a polynomial of degree c is uniquely determined by specifying an