Let x_1, x_2, ..., x_k be a system of orthonormal vectors in R^n. Consider the matrix A = x_1*x_1^T + ... + x_k*x_k^T. Let rank(M) and det(M) denote the rank and determinant of a matrix M respectively. Consider the following statements: I. rank(A) = det(A) = 1 II. rank(A) = det(A) = 0 Which of the following is/are TRUE? 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. Verify A is an orthogonal projection: A^T = (sum x_i*x_i^T)^T = sum x_i*x_i^T = A (symmetric). A^2 = sum_i sum_j x_i*(x_i^T*x_j)*x_j^T = sum_i x_i*x_i^T = A (idempotent). So A is an orthogonal projector.
  2. Statement A: Singular values = Eigenvalues: A is symmetric PSD (eigenvalues >= 0). Singular values of A = sqrt(eigenvalues of A^T*A) = sqrt(eigenvalues of A^2) = sqrt(eigenvalues of A) = eigenvalues of A. Statement A is TRUE.
  3. Statement B: Singular values in {0,1}: Since A is idempotent (A^2=A), eigenvalues satisfy lambda^2 = lambda, so lambda in {0,1}. Since singular values = eigenvalues, singular values are 0 or 1. Statement B is TRUE.
  4. Statements C and D: D (orthonormal => linearly independent) is TRUE by definition. C (LI => orthonormal) is FALSE: linearly independent vectors need not have unit norms or be mutually orthogonal.