Let p_1, p_2, ..., p_n be a system of orthonormal vectors in R^n. Consider the matrix A = p_1*p_1^T + ... + p_n*p_n^T. Which of the following statements is/are correct? A. Singular values of A are also its eigenvalues B. Singular values of A are either 0 or 1 C. Linearly independent vectors in R^n are orthonormal vectors D. Orthonormal vectors in R^n are linearly independent

GATE 2025 · Engineering Mathematics · Orthonormality · medium

Answer: Statements A and B are correct. Answer: A, B.

  1. Compute A for a complete ONB: If {p_1,...,p_n} is a complete orthonormal basis of R^n, then A = sum p_i*p_i^T = I (completeness relation). For k < n orthonormal vectors, A is an orthogonal projection of rank k with eigenvalues: k ones and (n-k) zeros.
  2. Check Statement A: singular values = eigenvalues: A is symmetric (A^T = A) and PSD. For such matrices, singular values = eigenvalues. A is TRUE.
  3. Check Statement B: singular values are 0 or 1: A = I_n (complete case) has eigenvalues all 1, so singular values = 1. More generally A is a projection so eigenvalues in {0,1}, hence singular values in {0,1}. B is TRUE.
  4. Check Statements C and D: D is TRUE: orthonormal vectors are always linearly independent (standard result). C is FALSE: linearly independent vectors need not be orthonormal (e.g., [1,0] and [1,1] are LI but not orthonormal).