is Hermitian, i.e., ¨© ¦ !
xB
£ ¤ ¤ ¤
@
¢
¦
¦
¨
¦ ¨© ¦
¨
©
¦ ¦
6@ 0@ 6 6@ 6@ 0 @B
—0
0 C ‘6© B
— C 0 B 0 C Y6
C
¤
is linear with respect to , i.e., ¦ ¨© ¦
xB
which satis¬es the following conditions:
¤ £
§ £§
¦
¨ ©
¨ ¦
¤
™ C g B
£¥ £ ¤ £
, into is any mapping from An inner product on a (complex) vector space
¡
¢ ˜{

! ¨ (
) % ¨©
§ $$§ )( ©¡ ! § § $ £ § "¨
$
¥ ¡ © ¥ ¡ ¥ © ¥ ¥
¥ ¨ ¤¤§ ¢¢¢
£ ¡© ¥¡
‘u @¥ £ ¡ ©§¢ ¥ ¥
u n ¨¦
¡ £ § ¡© £
u¢ n nd¥ ¥ ¡
§ ¢ £ ¢ ¡ © ¥ ¥
¡ ¥¡£¥
¡ §
j ‘ 1( C B G
‘ " $%%%"
$$
0
¨ "
¨
of is de¬ned by
PYQ y 1‘ ( & C x B t
0 ¨ ¢ §¥
¦
© ¦
¨ ¦ ¨
which is often rewritten in matrix notation as
t
x I C gB
¨ £ §¥
¦
"
¨¦ ¨ ¦
It is easy to verify that this mapping does indeed satisfy the three conditions required for
inner products, listed above. A fundamental property of the Euclidean inner product in
matrix computations is the simple relation
t
S™ x C I g % C x B
‘ ¨ ¤ §¥
B ¦
©¦
¨
¦ ¨ ¨©
¦
˜xB g B
The proof of this is straightforward. The adjoint of with respect to an arbitrary inner
C
˜ ©¦
¨ ¦ ¨ ¦ ¨
product is a matrix such that for all pairs of vectors and . A matrix
C
is selfadjoint, or Hermitian with respect to this inner product, if it is equal to its adjoint.
The following proposition is a consequence of the equality (1.5).
V¤ G S£
£¦
SRI FcP` `
Q
bQ
H QF ©U
T
Unitary matrices preserve the Euclidean inner product, i.e.,
g % C 0 x 0 B
B ¦ ¨ ¦
¨
C
0 ¦ ¨
for any unitary matrix and any vectors and .
£ g B C 1I 0 g B C 0 g 0 B
0
©¨¦
§ §¥ T ¦ ¨ "
¦ ¨ ¨© "
¦
Indeed, .
C
¦ ¤
¦
A vector norm on a vector space is a realvalued function on , which
£ £
satis¬es the following three conditions:
z (
h
7 a
7 a
7
¦ ¦ ¦ ¦
and iff .