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4 - S* -matrices

Published online by Cambridge University Press:  04 February 2010

Richard A. Brualdi
Affiliation:
University of Wisconsin, Madison
Bryan L. Shader
Affiliation:
University of Wyoming
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Summary

S*-matrices and SNS*-matrices

We first recall some basic facts from sections 1.2 and 2.1. Let A be an m by m + 1 matrix. By definition, A is an S*-matrix if and only if each submatrix of A of order m is an SNS-matrix. By (iv) of Theorem 2.1.1, A is an S*-matrix if and only if there exists A strict signing D such that AD and A(–D) are the only column signings of A each of whose rows are balanced. The matrix A is an S-matrix if and only if A is an S*-matrix and AIm+1 = A and A(–Iw+1) = –A are the only column signings of A each of whose rows are balanced.

Let v1, v2, … vm+1 be the column vectors of A. Then v1, v2,…, vm+1 are the vertices of an m -simplex whose interior contains the origin if and only if the right null space of A is spanned by A vector w each of whose entries is positive. This implies A geometric description of S-matrices [6]:

The matrix A is an S-matrix if and only if for each matrix à in Q(A), the column vectors of à are the vertices of an m-simplex containing the origin in its interior.

Clearly, each row of an S*-matrix must contain at least two nonzero entries. The following theorem [7] shows that an m by m + 1 matrix with at least three nonzero entries in each row is not an S*-matrix.

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Publisher: Cambridge University Press
Print publication year: 1995

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  • S* -matrices
  • Richard A. Brualdi, University of Wisconsin, Madison, Bryan L. Shader, University of Wyoming
  • Book: Matrices of Sign-Solvable Linear Systems
  • Online publication: 04 February 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511574733.005
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  • S* -matrices
  • Richard A. Brualdi, University of Wisconsin, Madison, Bryan L. Shader, University of Wyoming
  • Book: Matrices of Sign-Solvable Linear Systems
  • Online publication: 04 February 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511574733.005
Available formats
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Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • S* -matrices
  • Richard A. Brualdi, University of Wisconsin, Madison, Bryan L. Shader, University of Wyoming
  • Book: Matrices of Sign-Solvable Linear Systems
  • Online publication: 04 February 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511574733.005
Available formats
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