On complex eigenvalues of a real nonsymmetric matrix
Andy Wathen
TL;DR
The paper investigates representing real nonsymmetric matrices as products of two real symmetric matrices $B = T W$ and derives bounds linking the spectrum to the inertias of $T$ and $W$. It introduces a continuous-eigenvalue-trajectory argument via a parametric family $V(\theta)$ and leverages Jordan form to construct explicit factorisations, proving existence and non-uniqueness of such decompositions. A key result is that when $B$ has $m-s$ non-real eigenvalues, the numbers of positive eigenvalues of $T$ and $W$ are constrained by $\tfrac{1}{2}m-\tfrac{1}{2}s \le p \le \tfrac{1}{2}m+\tfrac{1}{2}s$ (with $s$ the number of real eigenvalues), and in the all-nonreal case the symmetric factors must have equal numbers of positive and negative eigenvalues. The methods also cover all-real spectra, where $B$ is similar to a real symmetric matrix and one can take $T$ positive definite, while non-real spectra require real Jordan blocks. Overall, the work clarifies spectral restrictions on symmetric-factor representations with implications for spectral theory and numerical factorisation of real matrices.
Abstract
We consider real non-symmetric matrices and their factorisation as a product of real symmetric matrices. The number of complex eigenvalues of the original matrix reveals restrictions on such factorisations as we shall prove.