Page 278 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes 2. As shown by example 2, in an arbitrary ordered basis , the relation between [T] and [T*]
 
is more complicated than that given in the corollary above.

Example 3: Let V be C , the space of complex n × 1 matrices, with inner product (X/Y) =
n×1
Y * X. If A is an n × n matrix with complex entries, the adjoint of the linear operator X AX is
the operator X A * X. For
(AX Y) = Y*AX = (A*Y)*X = (X A*Y)

Example 4: This is similar to Example 3. Let V be C n × n with the inner product (A B) =
tr (B*A). Let M be a fixed n × n matrix over C. The adjoint of left multiplication by M is left
multiplication by M*. Of course, ‘left multiplication by M’ is the linear operator L defined by
M
L (A) = MA.
M
(L (A) B) = tr (B* (MA))
M
= tr (MAB*)
= tr (AB*M)

= tr (A(M*B)*)
= (A L * (B)).
M
Thus (L )* = L . In the computation above, we twice used the characteristic property of the trace
M M*
function: tr (AB) = tr (BA).

Example 5: Let V be the space of polynomials over the field of complex numbers, with
the inner product.

1
(f g) = f ( ) ( ) .
t
dt
t
g
0
k
If f is a polynomial, f = a x , we let f a x k . That is, f is the polynomial whose associated
k k
polynomial function is the complex conjugate of that for f:
t
f ( ) = f ( ), t real
t
Consider the operator ‘multiplication by f,’ that is, the linear operator M defined by M (g) = fg.
f f
Then this operator has an adjoint, namely, multiplication by f For
.
(M(g) h) = (fg h)
f
1
t
g
h
t
t
= f ( ) ( ) ( )dt
0
1
= g ( )[ ( ) ( )]dt
t
t
t
f
h
0
= (g fh )
h
= (g M f ( ))
and so (M )* M .
f f
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