Question

Which ONE of the following statements is CORRECT regarding the eigenvalues of a real symmetric matrix?

• A. The eigenvalues are always complex
• B. The eigenvalues are always real
• C. The eigenvalues are always negative
• D. The eigenvalues are always equal to each other

Hint:

Recall the spectral theorem: real symmetric matrices always have real eigenvalues (and orthogonal eigenvectors), regardless of sign or repetition.

Answer 1

The Correct Option is B

Step 1: Recall the spectral theorem for real symmetric matrices. For any real symmetric matrix \(A\) (that is, \(A = A^T\) with all real entries), a fundamental result in linear algebra guarantees that all of its eigenvalues are real numbers, never complex, regardless of what the specific entries of the matrix are.

Step 2: Sketch why this is true. If \(\lambda\) is an eigenvalue with eigenvector \(v\) (possibly complex), then \(Av = \lambda v\). Taking the conjugate transpose and using \(A = A^T = \bar{A}\) (real entries), one can show \(\lambda = \bar\lambda\), which forces \(\lambda\) to be real.

Step 3: Eliminate the wrong options. Option (A) is false since real symmetric matrices specifically avoid complex eigenvalues (unlike general real non-symmetric matrices, which can have complex conjugate pairs). Option (C) is false because eigenvalues of a symmetric matrix can be positive, negative, or zero, depending on the definiteness of the matrix - only a negative-definite symmetric matrix has all-negative eigenvalues, which is not true for symmetric matrices in general. Option (D) is false since eigenvalues being all equal only happens for special matrices like scalar multiples of the identity, not for symmetric matrices in general.

Step 4: Conclude. The correct general statement is that eigenvalues of a real symmetric matrix are always real, which is option (B).