Decomposition formulas for Pfaffians
type: Item
instance of: scholarly article
title: Decomposition formulas for Pfaffians
zbMATH Open document ID: 0795.15004
author: Hema Srinivasan
published in: Journal of Algebra
publication date: 1994-09-11T00:00:00Z
review text: The Pfaffian of an alternating matrix \(X\) is the square root of the determinant of \(X\). If \(I=(i_ 1,i_ 2,\dots,i_ p)\) is a sequence of strictly increasing positive integers nor exceeding the order of \(X\), then the \(p\times p\) submatrix of \(X\), \(X(I)\), consisting precisely of the rows and columns in \(I\), is alternating and its Pfaffian is denoted by \(Pf(I)\). The signed Pfaffian of the matrix obtained from X after deleting the rows and columns in \(I\) is denoted by \(X_ I\). If \(I\) has \(t\) elements then \(\sum_ JX_{I,J}Pf(I,J)\) summed over all sequences \(J\) of length \(t\) corresponds to the expansion of the determinant by \(t\) rows in \(I\). The results of this paper give decompositions of \(Pf(I,J)\) and \(\sum_ JX_{I,J}Pf(J,K)\) for any sequences \(I,J\), and \(K\). In particular the Pfaffian of any given matrix can be expressed in terms of Pfaffian of lower orders.
reviewed by: A. P. Stone
Mathematics Subject Classification ID: 15A15
Mathematics Subject Classification ID: 15B57
Mathematics Subject Classification ID: 58A17
zbMATH DE Number: 554209
zbMATH Keywords: decomposition formulas
zbMATH Keywords: skew symmetric matrix
zbMATH Keywords: Laplace expansion
zbMATH Keywords: Pfaffian
zbMATH Keywords: alternating matrix
zbMATH Keywords: determinant
MaRDI profile type: Publication
full work available at URL: jabr.1994.1020
OpenAlex ID: W1994580746
DOI: 10.1006/JABR.1994.1020
description: scientific article; zbMATH DE number 554209