Question

All the definitions I have seen in the literature present the possibility of computing the $\epsilon$-pseudospectrum of a rectangular matrix $A^(m * n)$ with $m < n$ but I have not seen any definitions for the case $m > n$, why ? is there a trivial answer ?

Answer 1

I think that in the "short and fat" case (more columns than rows), the $\epsilon$-pseudospectrum with $\epsilon>0$ is the whole complex plane. So you don't get much information out of pseudospectra.

In more detail: Suppose you define the $\epsilon$-pseudospectrum of a "short and fat" matrix $A$ as the set of all complex numbers $z$ for which there is a vector $v$ and a matrix $E$ with $\|E\| < \epsilon$ such that $(A+E-zI)v = 0$ where $I$ is the rectangular "identity matrix", whose first $n$ columns form the $n$-by-$n$ square identity matrix and whose remaining columns are zero. In other words, $z$ is an "eigenvalue" of an $\epsilon$-perturbation of $A$. However, in the "short and fat" case, almost all $z$ are eigenvalues of the rectangular matrix $A$: if you decompose $A = [ \, A_1 \,\, A_2 \, ]$ where $A_1$ is square, and do the with $v$, then the eigenvalue equation $(A-zI)v = 0$ becomes $A_1v_1 + A_2v_2 = zv_1$, with solution $v_1 = -(A_1-zI)^{-1} A_2v_2$ if $z$ is not an eigenvalue of the square matrix $A_1$. I think that with arbitrarily small perturbations you can make every $z$ an eigenvalue of the rectangular matrix, and thus the $\epsilon$-pseudospectrum is the whole complex plane.