Some algebraic identities concerning determinants and permanents
type: Item
instance of: scholarly article
title: Some algebraic identities concerning determinants and permanents
zbMATH Open document ID: 0669.15005
DOI: 10.1016/0024-3795(89)90396-0
author: S. H. Smith
published in: Linear Algebra and its Applications
publication date: 1989-01-01T00:00:00Z
review text: The following identity is proved: \[ \det (X+A)=\sum_{\lambda \subseteq | n|}\sum_{\sigma \in D(\lambda)}(-1)^{| \lambda | +\iota (\sigma)}X_{\lambda c}per(A_{\sigma}), \] where A is an \(n\times n\) matrix, X is a diagonal matrix, D(\(\lambda)\) is the set of ordered partitions of \(\lambda\), \(\iota\) (\(\sigma)\) is the number of nonempty parts of \(\sigma\). The recurrent relation by \textit{T. Muir} [A relation between permanents of determinants. Proc. Roy. Soc. Edinburgh 22, 134-136 (1897)] follows from the above identity. Expansions for \(per(X+A)\), \(per^{-1}(I-XA)\), \(\det^{-1}(I-XA)\) are also demonstrated, where I denotes the identity matrix, \(\bar m=(m_ 1,m_ 2,...,m_ n)\) the nonnegative integer vector of n dimensions with coordinate sum \(| \bar m|\).
Mathematics Subject Classification ID: 15A15
zbMATH DE Number: 4095657
zbMATH Keywords: algebraic identities
zbMATH Keywords: determinants
zbMATH Keywords: permanents
zbMATH Keywords: recurrent relation
zbMATH Keywords: expansions
MaRDI profile type: Publication
full work available at URL: 0024-3795(89)90396-0
OpenAlex ID: W2118287793
cites work: Permanents
cites work: An identity involving permanents
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description: scientific article; zbMATH DE number 4095657