The GATE Grind

GATE 2023 CE (CE1) – Question 24

Engineering Mathematics · Linear Algebra: Matrix algebra, systems of linear equations, eigenvalues and eigenvectors · 1 mark · Multiple select

If $\mathbf{M}$ is an arbitrary real $n \times n$ matrix, then which of the following matrices will have non-negative eigenvalues?

  1. $\mathbf{M}^2$
  2. $\mathbf{M}\mathbf{M}^T$
  3. $\mathbf{M}^T\mathbf{M}$
  4. $(\mathbf{M}^T)^2$

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Correct answer: (B) $\mathbf{M}\mathbf{M}^T$; (C) $\mathbf{M}^T\mathbf{M}$

Explanation

The matrices $\mathbf{M}\mathbf{M}^T$ and $\mathbf{M}^T\mathbf{M}$ are symmetric and positive semi-definite, since $x^T\mathbf{M}\mathbf{M}^Tx = |\mathbf{M}^Tx|^2 \ge 0$, so all their eigenvalues are non-negative. The eigenvalues of $\mathbf{M}^2$ are the squares of those of $\mathbf{M}$, and these can be negative when $\mathbf{M}$ has complex eigenvalues (for example a rotation by 90°), so A and D do not always hold.