Alternating sign matrix: Difference between revisions

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In [[mathematics]], an '''alternating sign matrix''' is a [[square matrix]] of 0s, 1s, and −1s such that the sum of each row and column is 1 and the nonzero entries in each row and column alternate in sign. These matrices generalize [[Permutation matrix|permutation matrices]] and arise naturally when using [[Dodgson condensation]] to compute a determinant.{{cn|reason=This is not described in the linked article|date=April 2021}} They are also closely related to the [[six-vertex model]] with ___domain wall boundary conditions from [[statistical mechanics]]. They were first defined by William Mills, [[David P. Robbins|David Robbins]], and Howard Rumsey in the former context.

==ExampleExamples==

An example of an alternating sign matrix (that is not also a permutation matrix) is

A [[permutation matrix]] is an alternating sign matrix, and an alternating sign matrix is a permutation matrix if and only no entry equals {{math|−1}}.

[[File:Matrice signes alternants 4x4.svg|thumbnail|Puzzle picture]]

An example of an alternating sign matrix (that is not also a permutation matrix) is

[[File:Matrice signes alternants 4x4.svg|thumbnail|Puzzle picture]]

:<math>

\begin{bmatrix}