Linear-Time Computation of the Frobenius Normal Form for Symmetric Toeplitz Matrices via Graph-Theoretic Decomposition
Hojin Chu, Homoon Ryu
TL;DR
The paper tackles the computational bottleneck of obtaining the Frobenius normal form for symmetric Toeplitz matrices by rephrasing the problem as weighted Toeplitz graph component identification. It introduces α-type and β-type graph reductions that exploit Toeplitz regularity to enable a linear-time recursive decomposition, yielding the first row of each FNF diagonal block from the first row of A. The core contributions are (i) a linear-time graph-reduction framework, (ii) a reverse-reconstruction method for vertex components, and (iii) a linear-time scheme to assemble the FNF from the first rows of the blocks, extending algebraic structure to efficient symbolic computation. This approach demonstrates how structured combinatorial representations can achieve significant improvements over cubic-time matrix algorithms in symbolic linear algebra, with potential extensions to Hankel matrices and other Toeplitz-related families.
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 α-type reduction, which eliminates isolated vertices, and the β-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.