# On characteristic and permanent polynomials of a matrix

Special Matrices (2017)

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

top

## Abstract

top
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

top

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 -

## NotesEmbed?

top

You must be logged in to post comments.

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

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.