Question

Q6 5 Points Let (V, (,)) be an inner product space and T :V + V and S: V + V be self adjoint linear transformations. Show that To S:V + V is self adjoint if and only if S T =To S.

Answer 1

We have

$\langle Tv,w\rangle=\langle v,Tw\rangle,~~~~\langle Sv,w\rangle=\langle v,Sw\rangle$

for all $v,w\in V$ ; this is because are self-adjoint. Now,

$\begin{align*}&T\circ S~\mbox{is self adjoint}\\ \mbox{if and only if}\hspace{1cm}&\langle (T\circ S)v,w\rangle=\langle v,(T\circ S)w\rangle\hspace{1cm}\forall~v,w\in V\end{align*}$

But

$\begin{align*}\langle (T\circ S)v,w\rangle&=\langle T(Sv),w\rangle\\ &=\langle Sv,Tw\rangle\\ &=\langle v,(S\circ T)w\rangle\end{align*}$

Thus,

$\begin{align*}&T\circ S~\mbox{is self adjoint}\\ \mbox{if and only if}\hspace{1cm}&\langle (T\circ S)v,w\rangle=\langle v,(T\circ S)w\rangle\hspace{1cm}\forall~v,w\in V \\ \mbox{if and only if}\hspace{1cm}&\langle v,(S\circ T)w\rangle=\langle v,(T\circ S)w\rangle\hspace{1cm}\forall~v,w\in V \\ &T\circ S=S\circ T\end{align*}$