PAIRS OF MATRICES AND THEIR SIMULTANEOUS REDUCTION TO TRIANGULAR AND COMPANION FORMS IN THE CASE OF RANK(I — AZ) =1

H. Bart and H. K. Wimmer

No 272388, Econometric Institute Archives from Erasmus University Rotterdam

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.

Keywords: Agricultural and Food Policy; Research Methods/Statistical Methods (search for similar items in EconPapers)
Pages: 26
Date: 1990
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Persistent link: https://EconPapers.repec.org/RePEc:ags:eureia:272388

DOI: 10.22004/ag.econ.272388

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