A formula for the determinant of a sum of matrices
type: Item
instance of: scholarly article
title: A formula for the determinant of a sum of matrices
zbMATH Open document ID: 0628.15005
DOI: 10.1007/BF00401158
author: Christophe Reutenauer
author: Marcel Paul Schützenberger
published in: Letters in Mathematical Physics
publication date: 1987-01-01T00:00:00Z
review text: Given a square matrix \(x\) over a commutative ring, define functions \(\Lambda_ i(x)\) by the equality \[ \det (1-tx)=1-t\Lambda_ 1(x)+t^ 2\Lambda^ 2_ 2(x)+\ldots+(-1)^ nt^ n\Lambda^ n_ n(x)+\ldots . \] The following is the main theorem: Let \(X_ 1,...,X_ k\) be square matrices of the same order and \(n\geq 1\). Then \(\Lambda_ n(X_ 1+\ldots+X_ k)=\sum_{m}sgn(m)\Lambda (m)\) where the sum is extended to the \(k^ n\) monomials of length n of primitive circular words on \(X_ 1,\ldots,X_ k\). The Hamilton-Cayley theorem is deduced from this formula.
reviewed by: Yueh-er Kuo
Mathematics Subject Classification ID: 15A15
Mathematics Subject Classification ID: 15A24
zbMATH DE Number: 4021349
zbMATH Keywords: symmetric functions of the eigenvalues
zbMATH Keywords: determinant of a sum of matrices
zbMATH Keywords: commutative ring
zbMATH Keywords: circular words
zbMATH Keywords: Hamilton-Cayley theorem
MaRDI profile type: Publication
cites work: Factorizing words over an ordered alphabet
cites work: Q3659988
full work available at URL: https://doi.org/10.1007/bf00401158
OpenAlex ID: W2003518408
description: scientific article; zbMATH DE number 4021349