Question

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)

• A. All the elements of both matrices \( S_1 \) and \( S_2 \) are real.
• B. The matrix \( PQ \) can have complex eigenvalues.
• C. The matrix \( QP \) can have complex eigenvalues.
• D. The matrices \( P \) and \( Q \) commute.

Hint:

If two Hermitian matrices can be diagonalized by the same matrix, they commute. Also, the eigenvalues of Hermitian matrices are always real.

Answer 1

The Correct Option is A, D

We are given two Hermitian matrices \( P \) and \( Q \), and a matrix \( R \) that diagonalizes both of them. The matrices \( S_1 = RPR^{-1} \) and \( S_2 = RQR^{-1} \) are diagonal matrices. We are tasked with identifying the correct statements about these matrices. Step 1: Option (A) – Real Elements of \( S_1 \) and \( S_2 \)
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)