Page 262 - Advanced Linear Algebra

246 Advanced Linear Algebra

)
b Among the normal operators, the Hermitian operators are precisely
those for which all complex eigenvalues are real.
c Among the normal operators, the unitary operators are precisely those
)
for which all complex eigenvalues have norm .

2 )(Real case ) Let be a finite-dimensional real inner product space.
=
a ) B ²= ³ is normal if and only if
=~ ; p Ä p ; p > p Ä p >

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

basis for which
8
c
~ >

´µ 8 ?
)
b Among the real normal operators, the symmetric operators are those
in the decomposition of part 2a . )
for which there are no subspaces >
Hence, the following are equivalent for ²= ³ :
B

i) is symmetric.
ii is orthogonally diagonalizable.
)
)
iii has the orthogonal spectral resolution

b
~ b Ä
)
c Among the real normal operators, the orthogonal operators are
precisely those for which the eigenvalues are equal to f and the
(
)
)
described in part 2a have rows and columns of norm
matrices ´µ 8
, that is,
sin ccos
´µ ~ > 8 ?
cos sin

for some  . s
Proof. We have proved part 1a). As to part 1b), it is only necessary to look at a
diagonal matrix representing . This matrix has the eigenvalues of on its

(
main diagonal and so it is Hermitian if and only if the eigenvalues of are real.

Similarly, is unitary if and only if the eigenvalues of have absolute value
(

equal to .

We have proved part 2a). Parts 2b) and 2c) follow by looking at the matrix
(~ ´ µ 8 where 8 ~ 8 . This matrix is symmetric if and only if ( is diagonal,

and ( is orthogonal if and only if ~ f and the matrices ´ µ 8 have
orthonormal rows.
Matrix Versions
We can formulate matrix versions of the structure theorem for normal operators.