Linear-Time Computation of the Frobenius Normal Form for Symmetric Toeplitz Matrices via Graph-Theoretic Decomposition

TL;DR

This paper presents a linear-time algorithm for computing the Frobenius normal form of symmetric Toeplitz matrices by exploiting their graph-theoretic structure, significantly improving over previous cubic-time methods.

Contribution

The authors introduce a novel graph-theoretic approach with two reductions that enable linear-time decomposition and Frobenius normal form computation for symmetric Toeplitz matrices.

Findings

Achieved linear-time complexity for FNF computation of symmetric Toeplitz matrices.

Developed - and -type reductions for efficient graph decomposition.

Demonstrated structural regularities lead to computational efficiency in symbolic linear algebra.

Abstract

We introduce a linear-time algorithm for computing the Frobenius normal form (FNF) of symmetric Toeplitz matrices by utilizing their inherent structural properties through a graph-theoretic approach. Previous results of the authors established that the FNF of a symmetric Toeplitz matrix is explicitly represented as a direct sum of symmetric irreducible Toeplitz matrices, each corresponding to connected components in an associated weighted Toeplitz graph. Conventional matrix decomposition algorithms, such as Storjohann's method (1998), typically have cubic-time complexity. Moreover, standard graph component identification algorithms, such as breadth-first or depth-first search, operate linearly with respect to vertices and edges, translating to quadratic-time complexity solely in terms of vertices for dense graphs like weighted Toeplitz graphs. Our method uniquely leverages the structural regularities of weighted Toeplitz graphs, achieving linear-time complexity strictly with respect to vertices through two novel reductions: the \alpha-type reduction, which eliminates isolated vertices, and the \beta-type reduction, applying residue class contractions to achieve rapid structural simplifications while preserving component structure. These reductions facilitate an efficient recursive decomposition process that yields linear-time performance for both graph component identification and the resulting FNF computation. This work highlights how structured combinatorial representations can lead to significant computational gains in symbolic linear algebra.