# Is my rectangular matrix dissipative?

I'd like to find out if my rectangular matrix is a dissipative operator. Let me explain how the matrix is formed.

1. I'm given a set of parameter values $\lambda_i, i=1,2,...,N_d$ with $0\leq \lambda_i \leq 2\pi$. I'm also given a conditionally positive-definite radially symmetric function $\phi(r)$. I form a matrix $A_{i,j} = \phi\left(\sqrt{2 - 2\cos(\lambda_i - \lambda_j)}\right), i,j=1,2,...,N_d$. This matrix is circulant. It is guaranteed to be invertible if $\phi$ is conditionally positive-definite. $A$ is a parametric interpolation matrix used to form a parametric interpolant of cartesian positions. $\phi$ is infinitely smooth with global support.

2. I'm also given a set of parameter values $\lambda^s_i, i=1,2,...,N_s$ with $N_s >> N_d$ and $0 \leq \lambda^s_i \leq 2\pi$. I form a matrix $B_{i,j} = \phi\left(\sqrt{2 - 2\cos(\lambda^s_i - \lambda_j)}\right), i=1,2,...,N_s,j=1,2,...,N_d$. $B$ is a parametric evaluation matrix used to evaluate the parametric interpolant at $N_s$ locations in parameter space.

Given $A$ and $B$, I compute the matrix $E = BA^{-1}$. This matrix would always be applied to a vector of either the $X$ or $Y$ coordinates of a list of $N_d$ points in 2D.

How do I go about checking if $E$ is dissipative with respect to some norm? Would it involve checking the eigenvalues of $E^TE$ or the singular values of $E$ in some way? Is there even a way to do this for the vectors on which $E$ acts?

-
add comment