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Permutation-Like Matrices

Steven Robert Lippold

Abstract

Permutation Matrices are a well known class of matrices which encode the elements of the symmetric group on $d$ elements as a square $d\times d$ matrix. Motivated by [4], we define a similar class of matrices which are a generalization of Permutation Matrices. We give explicit formulas for the multiplication of these matrices. Lastly, we discuss the spectral radius, eigenvalues, and periodicity before giving a form of Birkhoff-Von Neumann's Theorem for Left Stochastic Matrices.

Permutation-Like Matrices

Abstract

Permutation Matrices are a well known class of matrices which encode the elements of the symmetric group on elements as a square matrix. Motivated by [4], we define a similar class of matrices which are a generalization of Permutation Matrices. We give explicit formulas for the multiplication of these matrices. Lastly, we discuss the spectral radius, eigenvalues, and periodicity before giving a form of Birkhoff-Von Neumann's Theorem for Left Stochastic Matrices.
Paper Structure (13 sections, 33 theorems, 53 equations)

This paper contains 13 sections, 33 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Permutation-Like Matrices
  3. Permutation-Like Matrices
  4. Symmetric Group Action on Permutation-Like Matrices
  5. Multiplication of Permutation-Like Matrices
  6. The Case $d=2$
  7. The Case $d=3$
  8. The Case $d>3$
  9. Properties of Permutation-Like Matrices
  10. Periodicity of Permutation-Like Matrices
  11. Eigenvalues of PLM
  12. Left Stochastic Matrices
  13. Remarks on Permutation-Like Matrices

Key Result

Proposition 2.1

$PL_d$ is a closed under matrix multiplication.

Theorems & Definitions (81)

  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Proposition 2.1
  • proof
  • Remark 2.6
  • Lemma 2.7
  • Remark 3.1
  • ...and 71 more