Algebra of L-banded Matrices

TL;DR

This paper explores the algebraic properties of L-banded matrices, a class of covariance matrices arising from optimized vector damping in iterative algorithms, providing explicit formulas and conditions for their key properties.

Contribution

It offers a comprehensive algebraic analysis of L-banded matrices, including explicit decompositions, determinants, and definiteness conditions, extending prior work on related matrix classes.

Findings

Derived explicit LDL and Cholesky decompositions.

Established conditions for matrix definiteness.

Provided formulas for determinants and inverses.

Abstract

Convergence is a crucial issue in iterative algorithms. Damping is commonly employed to ensure the convergence of iterative algorithms. The conventional ways of damping are scalar-wise, and either heuristic or empirical. Recently, an analytically optimized vector damping was proposed for memory message-passing (iterative) algorithms. As a result, it yields a special class of covariance matrices called L-banded matrices. In this paper, we show these matrices have broad algebraic properties arising from their L-banded structure. In particular, compact analytic expressions for the LDL decomposition, the Cholesky decomposition, the determinant after a column substitution, minors, and cofactors are derived. Furthermore, necessary and sufficient conditions for an L-banded matrix to be definite, a recurrence to obtain the characteristic polynomial, and some other properties are given. In addition, we give new derivations of the determinant and the inverse. (It's crucial to emphasize that some works have independently studied matrices with this special structure, named as L-matrices. Specifically, L-banded matrices are regarded as L-matrices with real and finite entries.)