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Linear complementarity properties of some classes of banded matrices

Samapti Pratihar, M. Seetharama Gowda, K. C. Sivakumar

Abstract

A banded matrix is a real square matrix where nonzero entries appear around the main diagonal. In this article, we consider linear complementarity properties of (variants) of banded matrices. Focusing on triangular matrices and the newly defined bidiagonal southwest matrices, we describe several results characterizing the Q-property in terms of the sign patterns and determinant of the given matrix. As a byproduct, we describe all Q-matrices of size 2 by 2. Extending these results to Euclidean Jordan algebras, we consider matrix-based linear transformations and study the Q-property. In particular, we show that a rank-one linear transformation of the form a\otimes b has the Q-property if and only if either a>0,b>0, or a<0, b<0.

Linear complementarity properties of some classes of banded matrices

Abstract

A banded matrix is a real square matrix where nonzero entries appear around the main diagonal. In this article, we consider linear complementarity properties of (variants) of banded matrices. Focusing on triangular matrices and the newly defined bidiagonal southwest matrices, we describe several results characterizing the Q-property in terms of the sign patterns and determinant of the given matrix. As a byproduct, we describe all Q-matrices of size 2 by 2. Extending these results to Euclidean Jordan algebras, we consider matrix-based linear transformations and study the Q-property. In particular, we show that a rank-one linear transformation of the form a\otimes b has the Q-property if and only if either a>0,b>0, or a<0, b<0.
Paper Structure (17 sections, 23 theorems, 72 equations)

This paper contains 17 sections, 23 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Linear complementarity problems
  4. Triangular matrices
  5. Bidiagonal southwest matrices
  6. Type-I bdsw matrices
  7. Type-II bdsw matrices
  8. Type-III bdsw matrices
  9. Type-IV bdsw matrices
  10. A characterization of ${\bf Q}$-matrices of size $2\times 2$
  11. Matrix-based linear transformations on a Euclidean Jordan algebra
  12. Euclidean Jordan algebras
  13. Symmetric cone LCP
  14. Matrix-based transformations
  15. Transformations corresponding to nonnegative and rank-one matrices
  16. ...and 2 more sections

Key Result

Theorem 2.1

Consider the LCP classes defined above. Then the following hold:

Theorems & Definitions (51)

  • Definition 2.1
  • Theorem 2.1
  • Theorem 2.2
  • proof
  • Proposition 2.1
  • proof
  • Theorem 3.1
  • proof
  • Example 3.1
  • Theorem 3.2
  • ...and 41 more