On characteristic and permanent polynomials of a matrix

Ranveer Singh; R. B. Bapat

Special Matrices (2017)

  • Volume: 5, Issue: 1, page 97-112
  • ISSN: 2300-7451

Abstract

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There is a digraph corresponding to every square matrix over ℂ. We generate a recurrence relation using the Laplace expansion to calculate the characteristic and the permanent polynomials of a square matrix. Solving this recurrence relation, we found that the characteristic and the permanent polynomials can be calculated in terms of the characteristic and the permanent polynomials of some specific induced subdigraphs of blocks in the digraph, respectively. Interestingly, these induced subdigraphs are vertex-disjoint and they partition the digraph. Similar to the characteristic and the permanent polynomials; the determinant and the permanent can also be calculated. Therefore, this article provides a combinatorial meaning of these useful quantities of the matrix theory. We conclude this article with a number of open problems which may be attempted for further research in this direction.

How to cite

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Ranveer Singh, and R. B. Bapat. "On characteristic and permanent polynomials of a matrix." Special Matrices 5.1 (2017): 97-112. <http://eudml.org/doc/288313>.

@article{RanveerSingh2017,
abstract = {There is a digraph corresponding to every square matrix over ℂ. We generate a recurrence relation using the Laplace expansion to calculate the characteristic and the permanent polynomials of a square matrix. Solving this recurrence relation, we found that the characteristic and the permanent polynomials can be calculated in terms of the characteristic and the permanent polynomials of some specific induced subdigraphs of blocks in the digraph, respectively. Interestingly, these induced subdigraphs are vertex-disjoint and they partition the digraph. Similar to the characteristic and the permanent polynomials; the determinant and the permanent can also be calculated. Therefore, this article provides a combinatorial meaning of these useful quantities of the matrix theory. We conclude this article with a number of open problems which may be attempted for further research in this direction.},
author = {Ranveer Singh, R. B. Bapat},
journal = {Special Matrices},
keywords = {Determinant; Permanent; Block; Block Graph; ℬ-partition},
language = {eng},
number = {1},
pages = {97-112},
title = {On characteristic and permanent polynomials of a matrix},
url = {http://eudml.org/doc/288313},
volume = {5},
year = {2017},
}

TY - JOUR
AU - Ranveer Singh
AU - R. B. Bapat
TI - On characteristic and permanent polynomials of a matrix
JO - Special Matrices
PY - 2017
VL - 5
IS - 1
SP - 97
EP - 112
AB - There is a digraph corresponding to every square matrix over ℂ. We generate a recurrence relation using the Laplace expansion to calculate the characteristic and the permanent polynomials of a square matrix. Solving this recurrence relation, we found that the characteristic and the permanent polynomials can be calculated in terms of the characteristic and the permanent polynomials of some specific induced subdigraphs of blocks in the digraph, respectively. Interestingly, these induced subdigraphs are vertex-disjoint and they partition the digraph. Similar to the characteristic and the permanent polynomials; the determinant and the permanent can also be calculated. Therefore, this article provides a combinatorial meaning of these useful quantities of the matrix theory. We conclude this article with a number of open problems which may be attempted for further research in this direction.
LA - eng
KW - Determinant; Permanent; Block; Block Graph; ℬ-partition
UR - http://eudml.org/doc/288313
ER -

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