Answer to Question #220155 in Linear Algebra for sabelo Bafana

Question #220155
Suppose S,TL(V) are self-adjoint. Prove that ST is self-adjoint if and only if ST=TS.
1
Expert's answer
2021-07-27T16:02:27-0400

Given S,T \isin L(V) are self-adjoint.

If suppose ST = TS,

    \implies ST + TS = 2(ST). Since 2(ST) is self-adjoint, where 2 is a real number.

consider, 2\langle ST(v), w \rangle = \langle 2ST(v), w \rangle = \langle v, 2ST(w) \rangle = 2\langle v, ST(w) \rangle

therefore, we get,

    \implies \langle ST(v), w \rangle = \langle v, ST(w) \rangle     \implies  ST is self-adjoint. 


Now suppose ST is self-adjoint. then \langle ST(v), w \rangle = \langle v, ST(w) \rangle

and \langle ST(v), w \rangle = \langle v,(ST)*w \rangle = \langle v, T*S*w \rangle = \langle v, TS(w) \rangle (since T, S are self-adjoint)

we get, \langle v, ST(w) \rangle = \langle v, TS(w) \rangle for all  v, w ∈ V

    \implies \langle v, (ST-TS)(w) \rangle = 0  for all v, w ∈ V ,

so setting v = (ST − T S)w

    \implies \langle (ST-TS)(w) , (ST-TS)(w) \rangle = 0

    \implies ||(ST − T S)w||2 = 0 for all w ∈ V ,

therefore,

ST − T S = 0      \implies ST = TS



Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment