Page 254 - Advanced Linear Algebra

238 Advanced Linear Algebra

=~ ; p Ä p ; p > p Ä p >

¹ is an indecomposable two-
where ¸Á à Á is the spectrum of and each >
dimensional -invariant subspace with an ordered basis for which
8

c
´µ ~ > 8 ?

Proof. We need only show that if has such a decomposition, then is normal.

=
But
! !
´µ ´µ ~ ² b ³0 ~ ´µ ´µ 8 8 8 8
is normal. It follows easily that is normal.

and so ´µ 8
Special Types of Normal Operators
We now want to introduce some special types of normal operators.

Definition Let be an inner product space.
=
(
1 ) B ²= ³ is self-adjoint also called Hermitian in the complex case and
symmetric in the real case if
)
i ~
(
2 ) B ²= ³ is skew self-adjoint also called skew-Hermitian in the
)
complex case and skew-symmetric in the real case if
i ~c
3 ) B ²= ³ is unitary in the complex case and orthogonal in the real case if
is invertible and

i ~ c
There are also matrix versions of these definitions, obtained simply by replacing
the operator by a matrix . Moreover, the operator is self-adjoint if and only

(

if any matrix that represents with respect to an ordered orthonormal basis is
E

self-adjoint. Similar statements hold for the other types of operators in the
previous definition.
In some sense, square complex matrices are a generalization of complex
numbers and the adjoint (conjugate transpose) is a generalization of the complex
conjugate. In looking for a better analogy, we could consider just the diagonal
matrices, but this is a bit too restrictive. The next logical choice is the set D of
normal matrices.

Indeed, among the complex numbers, there are some special subsets: the real
numbers, the positive numbers and the numbers on the unit circle. We will soon
see that a complex matrix is self-adjoint if and only if its complex eigenvalues
(