SOLUTION: Proof of Hermitian matrices: If A and B are Hermitian matrices, I need to show that BA = AB iff AB is Hermitian, but I can't figure out how...

Question 40403: Proof of Hermitian matrices: If A and B are Hermitian matrices, I need to show that BA = AB iff AB is Hermitian, but I can't figure out how...
Answer by venugopalramana(3286) (Show Source): You can put this solution on YOUR website!
A AND B ARE COMPLEX HERMITIAN MATRICES OF ORDER (N,N)
LET TRANSPOSE OF A AND B BE A' AND B'
SINCE A,B ARE HERMITIAN,WE HAVE
A=CONJUGATE (A') AND B= CONJ(B')��.I
CASE 1.
AB IS HERMITIAN.LET AB=C�..SO AB=C=CONJ(C')=CONJ(AB)'��..II
C'=(AB)'=B'A'
CONJ(C')=CONJ(B'A')=CONJ(B')CONJ(A')=BA�FROM I
BUT AB=CONJ(C') FROM II
SO AB=BA
CASE 2.
AB=BA= CONJ(B')CONJ(A')=CONJ(B'A')=CONJ(AB)'=AB�..
HENCE AB IS HERMITIAN.
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