Which of the following statements is/are TRUE for real matrices and their eigen-properties?
Options:
(A) Every real symmetric matrix has strictly real eigenvalues
(B) Eigenvectors corresponding to distinct eigenvalues of a real symmetric matrix are mutually orthogonal
(C) A real skew-symmetric matrix has eigenvalues that are either zero or purely imaginary
(D) If an $n \times n$ matrix has $n$ distinct eigenvalues, it is always diagonalizable
Answer Key:
A, B, C, D
Step-by-Step Solution & Explanation
1. **Spectral Theorem for Symmetric Matrices (Option A & B)**: Real symmetric matrices have strictly real eigenvalues and an orthonormal basis of eigenvectors. Eigenvectors of distinct eigenvalues satisfy $\langle v_i, v_j \rangle = 0$ (True).
2. **Skew-Symmetric Eigenvalues (Option C)**: If $A^T = -A$, let $Ax = \lambda x$. Then $\bar{x}^T A x = \lambda \|x\|^2$ and $(\bar{x}^T A x)^H = -\lambda \|x\|^2 = \bar{\lambda} \|x\|^2 \implies \bar{\lambda} = -\lambda$, so $\lambda$ is purely imaginary or zero (True).
3. **Distinct Eigenvalues (Option D)**: If all $n$ eigenvalues are distinct, their corresponding eigenvectors are linearly independent, ensuring an invertible modal matrix $P$ such that $P^{-1} A P = \Lambda$ (True).