Page 258 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes for all vectors and . Thus in the real case
1 2 1 2
( | ) = ... (3)
4 4
In the complex case we use (2) to obtain the more complicated expression

1 2 1 2 i 2 i 2
( | ) = i i ... (4)
4 4 4 4
Equations (3) and (4) are called the polarization identities. Note that (4) may also be written as
follows:

4
1 2
( | ) = i n i n .
4
n 1
The properties obtained above hold for any inner product on a real or complex vector space V,
regardless of its dimension. We turn now to the case in which V is finite-dimensional. As one
might guess, an inner product on a finite-dimensional space may always be described in terms
of an ordered basis by means of a matrix.

Suppose that V is finite-dimensional, that
= { , ...,  }
1 n
is an ordered basis for V, and that we are given a particular inner product on V; we shall show
that the inner product is completely determined by the values
G = ( | ) ... (5)
jk k j
it assumes on pairs of vectors in B. If = x and = y , then
k k j j
k j

( | ) = x |
n k
k

= x ( | )
k k
k
= x k y j ( k | j )
k j

= y G x
j jk k
, j k
= Y*GX

where X, Y are the coordinate matrices of , in ordered basis , and G is the matrix with entries
G = ( |a ). We call G the matrix of the inner product in the ordered basis . It follows from (5)
jk k j
that G is Hermitian i.e., that G = G*; however, G is a rather special kind of Hermitian matrix. For
G must satisfy the additional condition

X*GX > 0, X 0. … (6)
In particular, G must be invertible. For otherwise there exists an X 0 such that GX = 0, and for
any such X, (6) is impossible. More explicitly, (6) says that for any scalars x , ..., x not all of which
1 n
are 0.

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