Hi ALL !!!
First, apology my english !!
Look this problem:
A is a self adjoint operator such that tr(A^2) = 0. Proof that A = 0.
I thought use this property:
<A,B> = tr(B*A)
Because A is a self adjoint operator then I'll have that:
<A,A> = tr(A*A) = tr(AA) = tr(A^2) = 0 => <A,A> = 0 => A = 0.
But, I don't know how show that <A,B> = tr(B*A).
Some one have some idea ???
Thanks a lot !!
following up on previous post:
A is similar to diagonal matrix A' whose diagonal elements are the eigenvalues of A. A'^2 = 0 so the sum of the squares of the eigenvalues of A are zero so the eigenvalues of A are zero so A is zero. (Trace is an invariant).
EDIT: Argument invalid unless you show TrA^2=TrD^2
To finish off previous argument, if A and B are similar, Ax=Bx all x. So Ax=Dx=0 all x => A=0.
FrenandoRevilla took first step by pointing out a self-adjoint matrix is diagonalizable.
While I do like Fernando's solution, it's clearly the best, I do think your made a good observation. Namely, the map defines a norm (the Frobenius norm or Hilbert-Schmidt norm--the name's to taste) and so .
Neither you nor Fernando solved the problem, or even came close.
The solution is as I gave it, which I summarize here
Given self-adjoint A and TrA^2=0. Prove A=0.
1) A is self-adjoint therefore diagonalizable, D=PAP^-1 and similar to A.
2) D^2 = P(A^2)P^-1 therefore D^2 is similar to A^2 and their trace is equal.
3) TrD^2 = TrA^2 =0. Trace D^2 is sum of squares of eigenvalues so they are all 0 and D=0.
4) If A and B are similar Ax=Bx all B, so Ax = Dx = 0 all x, so A=0
Edited to change a typo in 4: "all x," not "all D." While I'm here, might as well improve step 4) which is correct but overly sophisticated. A = (P^-1)DP = 0.
Irrelevant Premise Proof: The Premise is obviously true but irrelevant. Ex:
1) The sky is blue
2) 3 is an integer
Therefore: sqrt2 is rational
Advanced Irrelative Premise Proof: The premises are unnecessarily obtuse. If carefully crafted, they can contain unproven assumptions or need not even be true.
These proofs are rock solid if an attempt is made to question the truth of the premise (the sky is blue).