A combinatorial approach to matrix algebra
type: Item
instance of: scholarly article
title: A combinatorial approach to matrix algebra
zbMATH Open document ID: 0609.05008
DOI: 10.1016/0012-365X(85)90192-X
published in: Discrete Mathematics
publication date: 1985-01-01T00:00:00Z
review text: In this paper, the author takes the combinatorial point of view that ''an \(n\times n\) matrix is the 'blueprint' of all possible edges one can draw on n given vertices, a determinant is the 'weight' of all permutation graphs and matrix-products represent paths''. After surveying such a combinatorial interpretation of matrix algebra, he presents new combinatorial proofs to five classical matrix identifies; namely, (1) MacMahon's master theorem, (2) the Cayley-Hamilton theorem, (3) the matrix tree theorem, (4) \(\det (AB)=(\det A)/\det B)\) and (5) \(\det (e^ A)=e^{tr(A)}\).
Mathematics Subject Classification ID: 05A99
Mathematics Subject Classification ID: 05C99
Mathematics Subject Classification ID: 15A24
zbMATH DE Number: 3985225
zbMATH Keywords: combinatorial matrix algebra
zbMATH Keywords: permutation graphs
zbMATH Keywords: paths
zbMATH Keywords: matrix identifies
author: Doron Zeilberger
reviewed by: Joel V. Brawley
MaRDI profile type: Publication
cites work: Q5775371
cites work: Combinatorial problems of commutation and rearrangements
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cites work: Q5343893
cites work: A Combinatorial Proof of Jacobi's Identity
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cites work: Line-digraphs, arborescences and theorems of Tutte and Knuth
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cites work: Q3937404
description: scientific article; zbMATH DE number 3985225