Eigenvalues of a matrix multiplied by its transpose I recall being told that the eigenvalues of the matrix formed by multiplying a matrix by its transpose are the squares of the eigenvalues of the original matrix. Is that true for any matrix? Surely not.

No they're definitely not. Here's an easy counterexample:
$$
A=\left( \begin{matrix}1 & 1 \\ 0 & 0 \end{matrix} \right)
$$
This matrix has eigenvalues $1$ and $0$ but 
$$
A^T A = \left( \begin{matrix}1 & 1 \\ 1 & 1 \end{matrix} \right)
$$
which has eigenvalues $2$ and $0$.
It is true if $A$ is a normal matrix (i.e. $A^T A = AA^T$) though.
