Let T be a linear operator on a complex inner product space V.
How to prove that T is Hermitian if and only if <T(x),x> is real for all x in V.
Dont really have any idea how to start.
Any help would be appreciated.
Thanks in advance.
Let T be a linear operator on a complex inner product space V.
How to prove that T is Hermitian if and only if <T(x),x> is real for all x in V.
Dont really have any idea how to start.
Any help would be appreciated.
Thanks in advance.
$\displaystyle T\,\,\,Hermitian\,\iff T=T^{*}\iff\,\,\forall,u\in V\,,\,\,<Tu,u>=<u,T^{*}u>=<u,Tu> $ $\displaystyle \iff <Tu,u>=<u,Tu>\,\,\,\forall u\in V$ ; but in
general, $\displaystyle <x,y>=\overline{<y,x>}$ , so:
$\displaystyle T$ is Hermitian $\displaystyle \iff \forall u\in V\,,\,\,<Tu,u>=<u,Tu>=\overline{<Tu,u>} \iff$ ... end the argument.
Tonio