Product of matrix transpose and matrix inverse Take any real matrix $A \in \mathbb{R}^{n \times n}$. We know that $A^TA$ has non-negative spectrum. Can we say anything interesting about the spectrum of $A^T A^{-1}$? If so, are there any references on the topic?
 A: Let $B=A^TA^{-1}$. Then $B^{-1}=A(A^T)^{-1}$ and hence $(B^{-1})^T=A^{-1}A^T$.
$B$ has the same complex spectrum as $(B^{-1})^T$ because $XY$ and $YX$ have identical characteristic polynomials for any two square matrices $X$ and $Y$ of the same sizes, where $X=A^T$ and $Y=A^{-1}$ in our case. $(B^{-1})^T$ has the same spectrum as $B^{-1}$ because every matrix has the same spectrum as its transpose.
Hence $B$ and $B^{-1}$ have the same spectrum and the eigenvalues of $B$, apart from $\pm1$, must occur in pairs of reciprocals $\{\lambda,\frac1\lambda\}$. Since $B$ is a real matrix, its non-real eigenvalues must occur in conjugate pairs too. In particular,


*

*if $\lambda$ is a non-real eigenvalue of $B$ whose modulus is not $1$, then $\lambda,\frac1\lambda,\overline{\lambda},\frac{1}{\overline{\lambda}}$ are four different eigenvalues having the same multiplicities in the spectrum of $B$;

*if $B$ is $(4n+2)\times(4n+2)$ and it has $4n$ non-real eigenvalues that do not lie on the unit circle, the remaining two eigenvalues must either be equal to $\pm1$, a pair of real numbers $\lambda,\frac1\lambda$, or a conjugate pair of complex eigenvalues $\lambda,\overline{\lambda}$ on the unit circle.

