Fast Symbolic Integer-Linear Spectra

TL;DR

This paper introduces a rapid symbolic eigenvalue solver for matrices with eigenvalues as integer-linear combinations of entries, enabling efficient high-dimensional symbolic analysis beyond numerical methods.

Contribution

It presents a novel fast symbolic eigenvalue solver and efficient generators for matrices with integer-linear eigenvalues, enhancing symbolic analysis capabilities.

Findings

Achieves faster eigenvalue computations for specific matrix classes.

Enables high-dimensional symbolic analysis where numerical methods struggle.

Provides tools for user interaction with linear operators.

Abstract

Here we contribute a fast symbolic eigenvalue solver for matrices whose eigenvalues are $\mathbb{Z}$-linear combinations of their entries, alongside efficient general and stochastic $M^{X}$ generators. Users can interact with a few degrees of freedom to create linear operators, making high-dimensional symbolic analysis feasible for when numerical analyses are insufficient.