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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Sign compatible expressions for minors of the matrix $I-A$
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by D. J. Hartfiel PDF
Proc. Amer. Math. Soc. 77 (1979), 304-308 Request permission

Abstract:

Let $A = ({a_{ij}})$ be an $n \times n$ nonnegative matrix having row sums less than or equal to one. This paper shows that the ijth minor of $I - A$ can be expressed as \[ {( - 1)^{i + j}}\sum {\Pi {r_k}{a_{pq}}} \] where \[ {r_k} = 1 - \sum \limits _{s = 1}^n {{a_{ks}}} \] and each $\Pi {r_k}{a_{pq}}$ is a product of exactly $n - 1$ numbers taken from ${r_k},{a_{pq}}$ for $k,p,q = 1, \ldots ,n$. This theorem is then used to obtain perturbation results concerning the matrix $I - A$.
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Additional Information
  • © Copyright 1979 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 77 (1979), 304-308
  • MSC: Primary 15A48; Secondary 15A45
  • DOI: https://doi.org/10.1090/S0002-9939-1979-0545585-9
  • MathSciNet review: 545585