# On characteristic and permanent polynomials of a matrix

Special Matrices (2017)

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

## Access Full Article

top## Abstract

top## How to cite

topRanveer 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 -

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.