Page 300 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes If we multiply the second equation by i and add the result to the first equation, we obtain
2f( , ) = 2f( , ).
Corollary: Let T be a linear operator on a complex finite-dimensional inner product space V.
Then T is self-adjoint if and only if (T | ) is real for every in V.
Theorem 4 (Principal Axis Theorem): For every Hermitian form f on a finite-dimensional inner
product space V, there is an orthonormal basis of V in which f is represented by a diagonal
matrix with real entries.
Proof: Let T be the linear operator such that f( , ) = (T | ) for all and in V. Then, since
f( , ) = ( , )f and (T | ) ( |T ) , it follows that

(T | ) = ( , ) ( |f T )
for all and ; hence T = T*. By Theorem 5 of unit 24, there is an orthonormal basis of V which
consists of characteristic vectors for T. Suppose { , ..., } is an orthonormal basis and that
1 n
T = c
j j j
for 1 j n. Then
f( , ) = (T | ) = c
k j k j kj k
and by Theorem 2 of unit 24 each c is real.
k
Corollary: Under the above conditions

( f x j j , y k k ) = c x y
j j j
j k j
Self Assessment

1. Which of the following functions f, defined on vectors = (x , x ) and (y , y ) = in c , are
2
1 2 1 2
sesquilinear forms on c 2
(a) f( , ) = (x – y ) + x y
2
1 1 2 2
(b) f( , ) = x y – x y
2 2 1
(c) f( , ) = x y
1 1
2. Let f be a non-degenerate form on a finite-dimensional space V. Show that each linear
operator S has an ‘adjoint’ relative to f’, i.e., an operator S’ such that f(S , ) = f( , S’ ) for
all , .

27.3 Summary

 In the introduction a review of the last units 24, 25, 26 is done. It is stated that the ideas
covered in these units are fundamental.

 In this unit forms on inner product space are studied and the relation between the forms
and the linear operator is established.
 A sesquilinear form is introduced and explained for all , , in the finite vector space V
and its relation with the linear operators.
 When the basis is an orthonormal basis, the matrix of the form f in is also matrix of the
linear transformation T .
fi

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