Eigenvalues of $A^TA$ Suppose $A$ is a $n\times n$ matrix in $M(\mathbb{R})$. I'd like to know if there is an exact formula for the eigenvalues of $A^TA$. Clearly, it's false that $\lambda^2$ is an eigenvalue of $A^TA$ if $\lambda$ is an eigenvalue of $A$.

Counterexample:
$$\begin{align*}
\operatorname{eig} \begin{pmatrix} 1 & 0 \\ 1 & 0 \end{pmatrix} &= \{0,1\} \\
\operatorname{eig} \begin{pmatrix} 1 & 1 \\ 0 & 0 \end{pmatrix}\begin{pmatrix} 1 & 0 \\ 1 & 0 \end{pmatrix} &= \{0,2\} \\
\end{align*}$$
