Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Algorithms for Recursive Block Matrices

Stephen M. Watt

TL;DR

This work develops and analyzes algorithms for recursive block matrices that preserve block structure in inversion and factorization tasks. It surveys block inversion results, including Schur‑complement based formulas and extensions to rings via Moore‑Penrose and Gonzalez‑Vega techniques, with a complexity view $T_{\mathrm{inv}}(n) \in O(T_\times(n))$. It then outlines a recursive LU decomposition framework for $2\times2$ block matrices, detailing conditions under which a block LU factorization $M=LU$ (up to permutation) can be computed without abandoning the block abstraction, and providing a thorough complexity analysis and strategies for handling singular blocks (including randomized approaches). The methods aim to enable efficient, structure‑preserving linear algebra on recursive block matrices in software, including noncommutative settings and rings with special properties, while offering practical pathways for implementation and performance estimation.

Abstract

We study certain linear algebra algorithms for recursive block matrices. This representation has useful practical and theoretical properties. We summarize some previous results for block matrix inversion and present some results on triangular decomposition of block matrices. The case of inverting matrices over a ring that is neither formally real nor formally complex was inspired by Gonzalez-Vega et al.

Algorithms for Recursive Block Matrices

TL;DR

This work develops and analyzes algorithms for recursive block matrices that preserve block structure in inversion and factorization tasks. It surveys block inversion results, including Schur‑complement based formulas and extensions to rings via Moore‑Penrose and Gonzalez‑Vega techniques, with a complexity view . It then outlines a recursive LU decomposition framework for block matrices, detailing conditions under which a block LU factorization (up to permutation) can be computed without abandoning the block abstraction, and providing a thorough complexity analysis and strategies for handling singular blocks (including randomized approaches). The methods aim to enable efficient, structure‑preserving linear algebra on recursive block matrices in software, including noncommutative settings and rings with special properties, while offering practical pathways for implementation and performance estimation.

Abstract

We study certain linear algebra algorithms for recursive block matrices. This representation has useful practical and theoretical properties. We summarize some previous results for block matrix inversion and present some results on triangular decomposition of block matrices. The case of inverting matrices over a ring that is neither formally real nor formally complex was inspired by Gonzalez-Vega et al.
Paper Structure (8 sections, 3 theorems, 18 equations)

This paper contains 8 sections, 3 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Inversion of Recursive Block Matrices
  3. LU Decomposition of Recursive Block Matrices
  4. Assuming a Nonsingular Block
  5. Block LU Decomposition
  6. LU Decomposition
  7. Complexity Analysis
  8. When All Blocks Are Singular

Key Result

Theorem 1

If $R$ is a formally real division ring and $M \in R^{n\times n}$ is invertible, then it is possible to compute $M^{-1}$ as $(M^TM)^{-1}M^T$ using only block operations.

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • Theorem 3