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Homework Statement
Let A and B be nxn positive self-adjoint matrices such that for all x [tex]\in[/tex] Cn, x*Ax = x*Bx. Prove that A = B. Equivalently, prove that if A, B are positive operators on H such that <Ax,x> = <Bx,x> [tex]\forall[/tex] x [tex]\in[/tex] H, then A = B. Hint: See Lemma 2.12.
Homework Equations
Lemma 2.12:
Let H be a Hilbert space over R or C and let T [tex]\rightarrow[/tex] H be a self-adjoint linear operator. Then we have:
(i) <Tx,x> is real for all x [tex]\in[/tex] H.
(ii) If H is over R, then for all x,y [tex]\in[/tex] H we have
<Tx,y> = 1/4 [<T(x+y),x+y> - <T(x-y),x-y>].
(iii) If H is over C, then for all x,y [tex]\in[/tex] H we have
<Tx,y> = 1/4 [<T(x+y),x+y> - <T(x-y),x-y>] + i/4 [<T(x+iy),x+iy> - <T(x-iy),x-iy>] .
The Attempt at a Solution
If we already know that <Ax,x> = <Bx,x>, doesn't it automatically follow that A = B? I'm not sure what has to be proven. I went through part (iii) of Lemma 2.12 and plugged in <Ax,x> for <Tx,y>, and everything canceled out so that <Ax,x> = <Ax,x>, but I don't think that this helped me at all. I'm just not sure what I'm being asked to prove...