Page 256 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes Since
2 2
( | ) = 2 x 2x y x
1 1 2 2
2
= x (x x )(x x )
1 1 2 1 2
It follows that ( | ) 0 if 0. Conditions (a), (b), and (c) of the definition are easily verified.
So ( | ) defines an inner product on F .
(2)
n n
Example 3: Let V be F , the space of all n × n matrices over F. Then V is isomorphic to
n2
F in a natural way. It follows from Example 1 that the equation
(A|B) = A B j , k
j k
, j k
defines an inner product on V. Furthermore, if we introduce the conjugate transpose matrix B*,
where B* = B we may express this inner product of F in terms of the trace function:
nn
jk j k
(A|B) = tr (A|B*) = tr (B* A).

For tr (AB*) = (AB *) jj
j

= A B * kj
jk
j k

= A B jk .
jk
j k

Example 4: Let F be the space of n  1 (column matrices over F, and let Q be an n  n
n1
invertible matrix over F. For X, Y in F n1 set
(X|Y) = Y*Q*QX.
We are identifying the 1 1 matrix on the right with its single entry. When Q is the identity
matrix, this inner product is essentially the same as that in Example 1; we call it the standard
inner product on F . The reader should note that the terminology ‘standard inner product’ is
n 1
used in two special contexts. For a general finite-dimensional vector space over F, there is no
obvious inner product that one may call standard.

Example 5: Let V be the vector space of all continuous complex-valued functions on the
unit interval, 0 t 1. Let

1
(f|g) = f ( ) ( ) .
g
t
t
dt
0
The reader is probably more familiar with the space of real-valued continuous functions on the
unit interval, and for this space the complex conjugate on g may be omitted.
Example 6: This is really a whole class of examples. One may construct new inner
products from a given one by the following method. Let V and W be vector spaces over F and

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