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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
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.

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

  • [1] 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).
  • [3] J. H. Wilkinson, The algebraic eigenvalue problem, Clarendon Press, Oxford, 1965. MR 0184422

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Keywords: Eigenvalues, characteristic polynomial, rank
Article copyright: © Copyright 1970 American Mathematical Society

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