Matrices

We are given two Hermitian matrices and , and a matrix that diagonalizes both of them. The matrices and are diagonal matrices. We are tasked with identifying the correct statements about these matrices. Step 1: Option (A) – Real Elements of and
Hermitian matrices have real eigenvalues, and since is an invertible matrix, the transformation and does not change the eigenvalues. Thus, all elements of and , which are the eigenvalues of and , are real. Therefore, Option (A) is correct. Step 2: Option (B) – Complex Eigenvalues of
The product of two Hermitian matrices and 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
Similar to Option (B), the matrix 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 and
If there exists a matrix that diagonalizes both and , it implies that and commute, i.e., . Therefore, Option (D) is correct. Final Answer: (A), (D)