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Proceedings of the American Mathematical Society

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

 
 

 

Factorization of singular matrices


Authors: A. R. Sourour and Kunikyo Tang
Journal: Proc. Amer. Math. Soc. 116 (1992), 629-634
MSC: Primary 15A23; Secondary 15A18
DOI: https://doi.org/10.1090/S0002-9939-1992-1097352-4
MathSciNet review: 1097352
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Abstract: We give a necessary and sufficient condition that a singular square matrix $ A$ over an arbitrary field can be written as a product of two matrices with prescribed eigenvalues. Except when $ A$ is a $ 2 \times 2$ nonzero nilpotent, the condition is that the number of zeros among the eigenvalues of the factors is not less than the nullity of $ A$. We use this result to prove results about products of hermitian and positive semidefinite matrices simplifying and strengthening some known results.


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

  • [1] C. K. Fong and A. R. Sourour, Sums and products of quasi-nilpotent operators, Proc. Roy. Soc. Edinburgh Sect. A 99 (1984), 193-200. MR 781097 (86g:47014)
  • [2] R. Horn and C. Johnson, Topics in matrix analysis, Cambridge Univ. Press, Cambridge, 1991. MR 1091716 (92e:15003)
  • [3] T. J. Laffey, Products of matrices, Generators and Relations in Groups and Geometries, NATO ASI Series, Kluwer, 1991. MR 1206912 (93m:15018)
  • [4] H. Radjavi, Products of hermitian matrices and symmetries, Proc. Amer. Math. Soc. 21 (1969), 369-372; Errata, Proc. Amer. Math. Soc. 26 (1970), 701. MR 0240116 (39:1470)
  • [5] A. R. Sourour, A factorization theorem for matrices, Linear and Multilinear Algebra 19 (1986), 141-147. MR 846549 (87j:15028)
  • [6] -, Nilpotent factorization of matrices, Linear and Multilinear Algebra (to appear). MR 1199060 (93k:15021)
  • [7] P. Y. Wu, Products of nilpotent matrices, Linear Algebra Appl. 96 (1987), 227-232. MR 910996 (88k:15015)
  • [8] -, Products of positive semidefinite matrices, Linear Algebra Appl. 111 (1988), 53-61. MR 974046 (90b:15010)
  • [9] -, The operator factorization problems, Linear Algebra Appl. 117 (1989), 35-63. MR 993030 (90g:47031)

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DOI: https://doi.org/10.1090/S0002-9939-1992-1097352-4
Article copyright: © Copyright 1992 American Mathematical Society

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