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Pairs Of Matrices And Their Simultaneous Reduction To Triangular And Companion Forms In The Case Of Rank(I — Az) =1

Author

Listed:
  • Bart, H.
  • Wimmer, H. K.

Abstract

This paper is concerned with simultaneous reduction to triangular and companion forms of pairs of matrices A, Z with rank( / — AZ) =1. Special attention is paid to the case where A is a first and Z is a third companion matrix. Two types of simultaneous triangularization problems are considered: ( ) the matrix A is to be transformed to upper triangular and Z to lower triangular form, ( 2) both A and Z are to be transformed to the same (upper) triangular form. The results on companions are made coordinate free by characterizing the pairs A, Z for which there exists an invertible matrix S such that S-1AS is of first and S-1ZS is of third companion type. One of the main theorems reads as follows: If rank(/ — AZ) =1 and saC 1 for every eigenvalue a of A and every eigenvalue of Z, then A and Z admit simultaneous reduction to complementary triangular forms.

Suggested Citation

  • Bart, H. & Wimmer, H. K., 1990. "Pairs Of Matrices And Their Simultaneous Reduction To Triangular And Companion Forms In The Case Of Rank(I — Az) =1," Econometric Institute Archives 272388, Erasmus University Rotterdam.
    • Handle: RePEc:ags:eureia:272388
    DOI: 10.22004/ag.econ.272388
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