Hermitian matrices P,Q are simultaneously diagonalized by the same invertible matrix R: RPR-1=S 1, RQR-1=S 2, where S 1,S 2 are diagonal. Which statements are correct?

$PQ$ can have complex eigenvalues.

$QP$ can have complex eigenvalues.

GATE 2021 PH – Question 53

Mathematical Physics · Matrices: similarity transformations, diagonalization, eigenvalues and eigen vectors · 2 marks · Multiple select

Hermitian matrices $P,Q$ are simultaneously diagonalized by the same invertible matrix $R$: $$RPR^{-1}=S_1,\qquad RQR^{-1}=S_2,$$ where $S_1,S_2$ are diagonal. Which statements are correct?

All elements of $S_1$ and $S_2$ are real.
$PQ$ can have complex eigenvalues.
$QP$ can have complex eigenvalues.
$P$ and $Q$ commute.

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Show answer and explanation

Correct answer: (A) All elements of $S_1$ and $S_2$ are real.; (D) $P$ and $Q$ commute.

Explanation

Similarity preserves eigenvalues and Hermitian matrices have real eigenvalues, so both diagonal matrices have real diagonal entries and zero off-diagonal entries: A holds. Diagonal matrices commute, so $R(PQ-QP)R^{-1}=S_1S_2-S_2S_1=0$, implying $PQ=QP$: D holds. Their products are similar to the real diagonal matrix $S_1S_2$, so their eigenvalues are real. B and C are false.

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