Matrices: a normal M, a projection Q, Hermitian transpose of product

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AI Thread Summary
The discussion focuses on establishing relationships between matrices, specifically proving QMQ=QM and QM*Q=QM*. The user initially expresses confusion about how to proceed with the problem involving Hermitian transposes. They later clarify their understanding, realizing that they can split the Hermitian transpose into its components and apply properties of transposition and complex conjugation. The user concludes that they have resolved their confusion and no longer need assistance. The thread highlights the importance of recognizing matrix properties in solving problems.
nomadreid
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Homework Statement
Given: orthogonal basis, normal matrix M, Projection P onto k-eigenspace, Q orthogonal complement of P, N*= Hermitian transpose of N. Prove: (QMQ)*=Q(M*)Q
Relevant Equations
M*M=MM*, PP=P, Q=(Id-P), Pv = w implies Mw=kw
Establish QMQ=QM and QM*Q = QM*, reducing the problem to
(QM)*=QM*
((Id-P)M)*=(Id-P)M*
(M-PM)*=M*-PM*
Applying to random vector v (ie. |v>),
(M-PM)*v = M*v-PM*v
Not sure where to go from here, although it is probably something that is supposed to be obvious.
Any help would be appreciated.
 
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This thread can be closed, as I understand how to do it now. Simply split the Hermitian transpose into its two parts as transposing then taking complex conjugate; use (AB)T= BTAT on ((QM)Q) twice, then use (this time using * as simply complex conjugate) (AB)*= A*B* twice. Sorry for the inconvenience; I should have seen this the first time.
 
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