Page 260 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes ( | )
= 2 , so ( | ) 0 and

( | ) ( | )
2
0 = 2 2

2
( | )( | ) 2 ( | )
= ( | ) 2 2

2 2 2
Hence ( | ) . Now using (iv) we find that
2 2 2
= ( | ) ( | )
2 2
= 2 Re( | )

2 2
2
2
= .
Thus, .
The inequality in (iii) is called the Cauchy-Schwarz inequality. It has a wide variety of applications.
The proof shows that if (for example) is non-zero, then ( | unless

( | )
= .
2

Thus, equality occurs in (iii) if and only if and are linearly dependent.

Example 7: If we apply the Cauchy-Schwarz inequality to the inner products given in
Examples 1, 3, and 5, we obtain the following:

1/2 1/2
2 2
(a) x y x y
k k k k
(b) tr (AB *) (tr (AA *)) 1/2 (tr (BB *)) 1/2

1 1 1/2 1 1/2
2
2
(c) f ( ) ( ) dx f ( ) dx g ( ) dx .
x
x
g
x
x
0 0 0
Definitions: Let and be vectors in an inner product space V. Then is orthogonal to if ( | )
= 0; since this implies is orthogonal to , we often simply say that and are orthogonal. If S
is a set of vectors in V, S is called an orthogonal set provided all pairs of distinct vectors in S are
orthogonal. An orthonormal set is an orthogonal set S with the additional property that 1 for
every in S.
The zero vector is orthogonal to every vector in V and is the only vector with this property. It is
appropriate to think of an orthonormal set as a set of mutually perpendicular vectors, each
having length 1.

254 LOVELY PROFESSIONAL UNIVERSITY