P and Q are two Hermitian matrices and there exists a matrix \( R \), which diagonalizes both of them, such that \( RPR^{-1} = S_1 \) and \( RQR^{-1} = S_2 \), where \( S_1 \) and \( S_2 \) are diagonal matrices. The correct statement(s) is(are)
Show Hint
- All the elements of both matrices \( S_1 \) and \( S_2 \) are real.
- The matrix \( PQ \) can have complex eigenvalues.
- The matrix \( QP \) can have complex eigenvalues.
- The matrices \( P \) and \( Q \) commute.
The Correct Option is A, D
Solution and Explanation
Hermitian matrices have real eigenvalues, and since \( R \) is an invertible matrix, the transformation \( RPR^{-1} = S_1 \) and \( RQR^{-1} = S_2 \) does not change the eigenvalues. Thus, all elements of \( S_1 \) and \( S_2 \), which are the eigenvalues of \( P \) and \( Q \), are real. Therefore, Option (A) is correct. Step 2: Option (B) – Complex Eigenvalues of \( PQ \)
The product of two Hermitian matrices \( P \) and \( Q \) can have complex eigenvalues. This is because the product of two matrices does not necessarily preserve the Hermitian property, and hence the eigenvalues can be complex. Thus, Option (B) is incorrect. Step 3: Option (C) – Complex Eigenvalues of \( QP \)
Similar to Option (B), the matrix \( QP \) does not necessarily have real eigenvalues since the product of two Hermitian matrices does not always result in a Hermitian matrix. Hence, Option (C) is incorrect. Step 4: Option (D) – Commutativity of \( P \) and \( Q \)
If there exists a matrix \( R \) that diagonalizes both \( P \) and \( Q \), it implies that \( P \) and \( Q \) commute, i.e., \( PQ = QP \). Therefore, Option (D) is correct. Final Answer: (A), (D)
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