Page 280 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes
based upon the observation that complex conjugation has the properties (z z ) = z z ,(z z )
1 2 1 2 1 2
,
= z z z = z. One must be careful to observe the reversal of order in a product, which the adjoint
1 2
operation imposes: (UT)* = T*U*. We shall mention extensions of this analogy as we continue
our study of linear operators on an inner product space. We might mention something along
these lines now. A complex number z is real if the only if z . z One might expect that the linear
operators T such that T = T* behave in some way like the real numbers. This is in fact the case. For
example, if T is a linear operator on a finite-dimensional complex inner product space, then
T = U + iU
1 2
where U = U* and U = U*. Thus, in some sense, T has a ‘real part’ and an ‘imaginary part.’ The
1 1 2 2
operators U and U satisfying U = U*, and U = U*, and are unique, and are given by
1 2 1 1 2 2
1
U = (T T *)
1 2
1
U = (T T *).
2 2i
A linear operator T such that T = T* is called self-adjoint (for Hermitian). If  is an orthonormal
basis for V, then

[T*] = [T]*
 
and so T is self-adjoint if and only if its matrix in every orthonormal basis is a self-adjoint
matrix. Self-adjoint operators are important, not simply because they provide us with some sort
of real and imaginary part for the general linear operator, but for the following reasons:
(1) Self-adjoint operators have many special properties. For example, for such an operator there
is an orthonormal basis of characteristic vectors. (2) Many operators which arise in practice are
self-adjoint. We shall consider the special properties of self-adjoint operators later.

Self Assessment

1. Let V be a finite-dimensional inner product space T a linear operator on V. If T is invertible,
show that T* is invertible and (T*) = (T )*.
–1
–1
2. Show that the product of two self-adjoint operators is self-adjoint if any only if the two
operators commute.

25.3 Summary

 The linear functional f concept is also a form of inner product on a finite-dimensional
inner product space.
 The fact that f has the form f( ) = ( ) for some in V helps us to prove the existence of the
‘adjoint’ of a linear operator T on V.

 A linear operator T such that T = T* is called self-adjoint (or Hermitian) and so T is
self- adjoint if and only if its matrix in every orthonormal basis is a self-adjoint matrix.

25.4 Keywords

A linear functional f on a finite dimensional inner product space is ‘inner-product with a fixed
vector in the space’. Let be some fixed vector in any inner product space V, we then define a
function f from V into the scalar field by

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