Page 212 - Advanced Linear Algebra

196 Advanced Linear Algebra

is almost upper triangular. Hence, if
0 c
8~ > ?
7
block
then is invertible and
8

8´(µ 8 ~ > 0 c c ? > ( i ? > 0 c c ? ~ > ( i ?
8
7 ( 7 <
is almost upper triangular.
Unitary Triangularizability
Although we have not yet discussed inner product spaces and orthonormal
bases, the reader may very well be familiar with these concepts. For those who
are, we mention that when is a real or complex inner product space, then if an
=

operator on = can be triangularized (or almost triangularized) using an
ordered basis , it can also be triangularized (or almost triangularized) using an
8
orthonormal ordered basis .
E
To see this, suppose we apply the Gram–Schmidt orthogonalization process to a
basis 8 that triangularizes (or almost triangularizes) . The
~²# Á Ã Á # ³
resulting ordered orthonormal basis E ~²" Á Ã Á " ³ has the property that

º# ÁÃÁ# » ~ º" ÁÃÁ" »

for all  . Since ´ µ 8 is (almost) upper triangular, that is,
#  º# ÁÃÁ# »

for all  , it follows that
"  º # ÁÃÁ # » ‹ º# ÁÃÁ# » ~ º" ÁÃÁ" »

is also (almost) upper triangular.
and so the matrix ´µ E
A linear operator is unitarily upper triangularizable if there is an ordered

orthonormal basis with respect to which is upper triangular. Accordingly,

when = is an inner product space, we can replace the term “upper
triangularizable” with “unitarily upper triangularizable” in Schur's theorem. (A
similar statement holds for almost upper triangular matrices.)
Diagonalizable Operators
Definition A linear operator ²= ³ is diagonalizable if there is an ordered

B
basis 8 of = for which the matrix ~²# Á Ã Á # ³ ´ µ 8 is diagonal, or
equivalently, if