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Proceedings of the American Mathematical Society

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Sign compatible expressions for minors of the matrix $ I-A$


Author: D. J. Hartfiel
Journal: Proc. Amer. Math. Soc. 77 (1979), 304-308
MSC: Primary 15A48; Secondary 15A45
MathSciNet review: 545585
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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

$\displaystyle {( - 1)^{i + j}}\sum {\Pi {r_k}{a_{pq}}} $

where

$\displaystyle {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$.

References [Enhancements On Off] (What's this?)

  • [1] F. R. Gantmacher, The theory of matrices. Vol. 2, Chelsea, New York, 1960.
  • [2] Sailes Kumar Sengupta, Comparison of eigenvectors of irreducible stochastic matrices, Linear Algebra and Appl. 12 (1975), no. 2, 101–110. MR 0382313

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1979-0545585-9
Article copyright: © Copyright 1979 American Mathematical Society