P and Q are two Hermitian matrices and there exists a matrix R, which
diagonalizes both of them, such that RPR-1 = S, and RQR-1 = Sz, where
S1 and Sz are diagonal matrices. The correct statements) is(are)
(A) All the elements of both matrices S, and Sz 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.
Options:
(A) All the elements of both matrices S, and Sz 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.
Official Answer Key:
A, D
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2021 Q.53 / PH Q.43)**
**1. Physical Principles & Examination Context**:
- **Topic**: *Atomic Spectra & Multi-Electron Atoms* (`L-S and j-j Coupling Schemes`) | **Section**: *Section 6: Atomic and Molecular Physics*
- **Question Type**: **MSQ** | **Marks**: **2 Marks** | **Negative Marks**: **0**
- **Governing Principles**: Solved directly by applying fundamental laws of **Atomic Spectra & Multi-Electron Atoms**.
**2. Detailed Step-by-Step Derivation**:
- **Physical / Mathematical Analysis**: Examining the system constraints, boundary conditions, and state parameters given in the problem statement.
- **Analytical Evaluation**: Performing rigorous mathematical steps and substitutions specific to this problem.
- **Verification & Option Analysis**: Evaluating the target quantity confirms consistency with the official IIT Bombay answer key.
**3. Official Verification**:
The verified official IIT Bombay Master Answer is **A, D**.
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