Eigenvalues of $A^{T}A$

Question

Let $\lambda_{i}(M)$ denote the $i$th eigenvalue of the square matrix $M$, and $T$ denote the matrix transpose.

Is it true that $|\lambda_{i}(A^{T}A)|=|\lambda_{i}(A)|^{2}$ for every square matrix $A$?

Thank you very much!

Answer 1

No. Take $A=\begin{bmatrix}0 & 1 \\ 0 & 0 \end{bmatrix}$. The spectral radius is zero, but $A^TA=\begin{bmatrix}0 & 0 \\ 0 & 1 \end{bmatrix}$ has an eigenvalue at one.

Answer 2

The square roots of the eigenvalues of $A^TA$ are also called "singular values" of $A$, and have no relationship to the eigenvalues of $A$, in general. The only exception is when $A$ is symmetric (or more general a normal matrix). Then your relation holds. When $A$ is even symmetric (or Hermitian) and positive definite the eigenvalues and the singular values coincide.