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Reducing the rank of $ (A-\lambda B)$


Authors: Gerald L. Thompson and Roman L. Weil
Journal: Proc. Amer. Math. Soc. 26 (1970), 548-554
MSC: Primary 65.40
DOI: https://doi.org/10.1090/S0002-9939-1970-0266418-6
MathSciNet review: 0266418
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Abstract: The rank of the $ n \times n$ matrix $ (A - \lambda I)$ is $ n - J(\lambda )$ when $ \lambda $ is an eigenvalue occurring in $ J(\lambda ) \geqq 0$ Jordan blocks of the Jordan normal form of $ A$. In our principal theorem we derive an analogous expression for the rank of $ (A - \lambda B)$ for general, $ m \times n$, matrices. When $ J(\lambda ) > 0,\lambda $ is a rank-reducing number of $ (A - \lambda I)$. We show how the rank-reducing properties of eigenvalues can be extended to $ m \times n$ matrix expressions $ (A - \lambda B)$. In particular we give a constructive way of deriving a polynomial $ P(\lambda ,A,B)$ whose roots are the only rank-reducing numbers of $ (A - \lambda B)$. We name this polynomial the characteristic polynomial of $ A$ relative to $ B$ and justify that name.


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  • [1] F. R. Gantmacher, Matrizenrechnung. II. Spezielle Fragen und Anwendungen, Hochschulbücher für Mathematik, Bd. 37, VEB Deutscher Verlag der Wissenschaften, Berlin, 1959 (German). MR 0107647
    F. R. Gantmacher, Applications of the theory of matrices, Translated by J. L. Brenner, with the assistance of D. W. Bushaw and S. Evanusa, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1959. MR 0107648
    F. R. Gantmacher, The theory of matrices. Vols. 1, 2, Translated by K. A. Hirsch, Chelsea Publishing Co., New York, 1959. MR 0107649
  • [2] Gerald L. Thompson and Roman L. Weil, The roots of matrix pencils $ (Ay = \lambda By)$: Existence, calculations, and relations to game theory, Management Sciences Research Report No. 172, Graduate School of Industrial Administration, Carnegie-Mellon University, Report #6936 of the Center for Mathematical Studies in Business and Economics of the University of Chicago, Chicago, Ill., 1969; Linear Algebra and Appl. (submitted).
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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1970-0266418-6
Keywords: Eigenvalues, characteristic polynomial, rank
Article copyright: © Copyright 1970 American Mathematical Society