Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Tridiagonal and single-pair matrices and the inverse sum of two single-pair matrices

Sebastien Bossu

Abstract

A novel factorization for the sum of two single-pair matrices is established as product of lower-triangular, tridiagonal, and upper-triangular matrices, leading to semi-closed-form formulas for tridiagonal matrix inversion. Subsequent factorizations are established, leading to semi-closed-form formulas for the inverse sum of two single-pair matrices. An application to derive the symbolic inverse of a particular Gram matrix is presented, and the numerical stability of the formulas is studied.

Tridiagonal and single-pair matrices and the inverse sum of two single-pair matrices

Abstract

A novel factorization for the sum of two single-pair matrices is established as product of lower-triangular, tridiagonal, and upper-triangular matrices, leading to semi-closed-form formulas for tridiagonal matrix inversion. Subsequent factorizations are established, leading to semi-closed-form formulas for the inverse sum of two single-pair matrices. An application to derive the symbolic inverse of a particular Gram matrix is presented, and the numerical stability of the formulas is studied.
Paper Structure (16 sections, 11 theorems, 56 equations, 3 figures, 1 table, 1 algorithm)

This paper contains 16 sections, 11 theorems, 56 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction and motivation
  2. Single-pair matrices
  3. The inverse of a single-pair matrix is a tridiagonal matrix
  4. The inverse of a symmetric tridiagonal matrix may be written as a single-pair matrix
  5. A single-pair matrix is "vector-bihomogeneous"
  6. Problem
  7. Main result and organization of this paper
  8. Tridiagonal decomposition of the sum of two single-pair matrices
  9. Single-pair decomposition of the sum of two single-pair matrices
  10. Application: symbolic inverse of the Gram matrix of a system of ramp functions
  11. Factorizations
  12. Symbolic inverses
  13. Continuant sequence
  14. Application: numerical inverse of the sum of two single-pair matrices
  15. Numerical stability for a family of matrices with determinant $-\varepsilon/9$
  16. ...and 1 more sections

Key Result

Lemma 1

If $\mathbb L \coloneqq (l_{i,j}),\mathbb T \coloneqq (t_{i,j}), \mathbb U \coloneqq (u_{i,j})$ are respectively lower-triangular, tridiagonal and upper-triangular square matrices of order $n$, their product is an $n\times n$ matrix with coefficients where any void sum is deemed to be zero. Similarly,

Figures (3)

  • Figure 1: MAE of matrices with determinant $-\varepsilon/9$ as a function of $\varepsilon\in[10^{-6},0.1]$
  • Figure 2: MAE distribution charts, 27mn. matrices with spectrum $\{-1, \varepsilon, 1\}$.
  • Figure 3: MAE scatter, 27mn. matrices with spectrum $\{-1, \varepsilon, 1\}$.

Theorems & Definitions (23)

  • Definition
  • Remark
  • Lemma 1
  • Theorem 1
  • Remark 1.1
  • Remark 1.2
  • Corollary 1.1
  • Theorem 2
  • Remark 2.1
  • Remark 2.2
  • ...and 13 more