On characteristic and permanent polynomials of a matrix

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 polynom...

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Bibliographic Details
Published in:Special matrices 2017-01, Vol.5 (1), p.97-112
Main Authors: Singh, Ranveer, Bapat, R. B.
Format: Article
Language:English
Subjects:
05C20
15A15
68R10
Block
Block Graph
Determinant
Permanent
ℬ-partition
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Summary: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.
ISSN:2300-7451
2300-7451
DOI:10.1515/spma-2017-0010