Let $A$ be an $n \times n$ real symmetric matrix. Which of the following statements is/are ALWAYS TRUE?
Options:
(A) All eigenvalues of $A$ are real numbers
(B) Eigenvectors corresponding to distinct eigenvalues are mutually orthogonal
(C) The matrix $A$ is diagonalizable using an orthogonal modal matrix
(D) If all eigenvalues of $A$ are positive, then $A$ is positive definite
Answer Key:
A, B, C, D
Step-by-Step Solution & Explanation
For any real symmetric matrix $A$:
1. All eigenvalues are strictly real. (A is true)
2. Eigenvectors belonging to distinct eigenvalues satisfy $(v_i, v_j) = 0$, hence mutually orthogonal. (B is true)
3. By the Spectral Theorem, there exists an orthogonal matrix $P$ such that $P^T A P = D$. (C is true)
4. A real symmetric matrix with strictly positive eigenvalues satisfies $x^T A x > 0$ for all $x \neq 0$, which is the definition of positive definiteness. (D is true)
Hence, all choices A, B, C, and D are correct.