Question

Let \( x_1, x_2, x_3, x_4, x_5 \) be a system of orthonormal vectors in \( \mathbb{R}^{10} \). Consider the matrix \[ A = x_1 x_1^T + x_2 x_2^T + x_3 x_3^T + x_4 x_4^T + x_5 x_5^T. \] 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. Determinant of A is 1
• D. A is invertible

Hint:

In an orthonormal basis, the sum of rank-1 matrices formed by outer products of orthonormal vectors results in a projection matrix. The singular values of a symmetric matrix are equal to its eigenvalues.

Answer 1

The Correct Option is A, B

We are given that \( A = x_1 x_1^T + x_2 x_2^T + x_3 x_3^T + x_4 x_4^T + x_5 x_5^T \), where \( x_1, x_2, x_3, x_4, x_5 \) are orthonormal vectors in \( \mathbb{R}^{10} \). The matrix \( A \) is a projection matrix onto the 5-dimensional subspace spanned by these vectors, which implies:

The eigenvalues of \( A \) are 1 (with multiplicity 5) and 0 (with multiplicity 5).
The singular values of \( A \) are the square roots of the eigenvalues of \( A \), so the singular values are either 0 or 1.

Thus, Option (A) is correct, and Option (B) is also correct.

The determinant of \( A \) is 0 because it has eigenvalues 0, and \( A \) is not invertible, so Options (C) and (D) are incorrect.