Question

If \(P\) and \(Q\) are Hermitian matrices, which of the following is/are true?
(A matrix \(P\) is Hermitian if \(P = P^{\dagger}\), where the elements \(p_{ij}^{\dagger} = p_{ji}^{*}\))

• A. \(PQ + QP\) is always Hermitian
• B. \(i(PQ - QP)\) is always Hermitian
• C. \(PQ\) is always Hermitian
• D. \(PQ - QP\) is always Hermitian

Hint:

The sum of two Hermitian matrices or any expression symmetric under Hermitian conjugation remains Hermitian, but their product generally does not.

Answer 1

The Correct Option is A, B

Step 1: Recall the property of Hermitian matrices.
A matrix \(A\) is Hermitian if \(A^{\dagger} = A\). For two Hermitian matrices \(P\) and \(Q\), we have \(P^{\dagger} = P\) and \(Q^{\dagger} = Q\).

Step 2: Check \(PQ + QP\).
\[ (PQ + QP)^{\dagger} = Q^{\dagger} P^{\dagger} + P^{\dagger} Q^{\dagger} = QP + PQ = PQ + QP \] Thus, \(PQ + QP\) is Hermitian.

Step 3: Check \(i(PQ - QP)\).
\[ [i(PQ - QP)]^{\dagger} = -i(Q^{\dagger} P^{\dagger} - P^{\dagger} Q^{\dagger}) = i(PQ - QP) \] Hence, \(i(PQ - QP)\) is Hermitian.

Step 4: Check the remaining options.
- \(PQ\) is not necessarily Hermitian because \((PQ)^{\dagger} = QP\). - \(PQ - QP\) is anti-Hermitian since \((PQ - QP)^{\dagger} = -(PQ - QP)\).

Step 5: Conclusion.
Hence, the correct statements are (A) and (B).