Page 299 - DMTH502_LINEAR_ALGEBRA

Unit 27: Introduction and Forms on Inner Product Spaces

The matrix of f in another basis Notes
n
= P ij i , (1 j n)
j
i 1
is given by the equation

A = P*AP. (2)
For
A = f( , )
jk k j

= f P sk s , P rj r
s r

= P A P
rj rs sk
, r s
= (P*AP) .
jk
–1
Since P *= P for unitary matrices, it follows from (2) that results concerning unitary equivalence
may be applied to the study of forms.
Theorem 2: Let f be a form on a finite-dimensional complex inner product space V. Then there is
an orthonormal basis for V in which the matrix of f is upper-triangular.

Proof: Let T be the linear operator on V such that f( , ) = (T | ) for all and . By Theorem 12
of unit 26 there is an orthonormal basis ( , ..., ) in which the matrix of T is upper-triangular.
1 n
Hence.
f( , ) = (T | ) = 0
k j k j
when j > k.
Definition: A form f on a real or complex vector space V is called Hermitian if

f( , ) = ( , )f
for all and in V.
If T is a linear operator on a finite-dimensional inner product space V and f is the form
f( , ) = (T | )

then ( , ) ( |f T ) ( * | ) ; so f is Hermitian if and only if T is self-adjoint.
b
T
When f is Hermitian f( , ) is real for every , and on complex spaces this property characterizes
Hermitian forms.
Theorem 3: Let V be a complex vector space and f a form on V such that f( , ) is real for every .
Then f is Hermitian.
Proof: Let and be vectors in V. We must show that f( , ) = ( , )f . Now

f( + , + ) = f( , ) + f( , ) + f( , ) + f( , ).
Since f( + , + ) = f( , ), and f( , ) are real, the number f( , ) + f( , ) is real. Looking at the
same argument with + i instead of + , we see that – if ( , ) + if ( , ) is real. Having
concluded that two numbers are real, we set them equal to their complex conjugates and obtain
f( , ) + f( , ) = ( , )f + ( , )f

–if( , ) + if ( , ) = ( , )if – ( , )if

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