# Prove if T is Hermitian, then <T(x), x> is real for alll x in V.

• Apr 18th 2009, 07:25 AM
ericmik
Prove if T is Hermitian, then <T(x), x> is real for alll x in V.
Let T be a linear operator on a complex inner product space V.

1.Prove if T is Hermitian, then <T(x), x> is real for alll x in V.

2. Prove that if <T(x), x> is real for all x in V, then T is Hermitian.

I'd be grateful with any degree of help!

thanks!
(Clapping)
• Apr 18th 2009, 07:44 AM
ericmik
Quote:

Originally Posted by ericmik
Let T be a linear operator on a complex inner product space V.

1.Prove if T is Hermitian, then <T(x), x> is real for alll x in V.

2. Prove that if <T(x), x> is real for all x in V, then T is Hermitian.

I'd be grateful with any degree of help!

thanks!
(Clapping)

i sorta came up with a solution here,

T = T^*

then<T(x), x> = <x, T(x)>
by the definition of inner product space: <T(x), x> = conjugate(<x, T(x)>)

thus <x, T(x)> = conjugate<x, T(x)>

but is it sufficient to say <x, T(x)> is real??
• Apr 18th 2009, 08:30 AM
Moo
Quote:

Originally Posted by ericmik
i sorta came up with a solution here,

T = T^*

then<T(x), x> = <x, T(x)>
by the definition of inner product space: <T(x), x> = conjugate(<x, T(x)>)

thus <x, T(x)> = conjugate<x, T(x)>

but is it sufficient to say <x, T(x)> is real??

Consider <x,T(x)>=Re(<x,T(x)>)+i*Im(<x,T(x)>)
Conjugate of <x,T(x)>=Re(<x,T(x)>)-i*Im(<x,T(x)>)
Now what if you add these two equations ?