Can the TPRI structure help us to solve the algebraic eigenproblem?

Authors:
Victor Y. Pan
Affiliations:
University of New York, Bronx, NY
Venue:
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Year:
2005

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Abstract

We modify the customary approach to solving the algebraic eigenproblem. Instead of applying the QR algorithm to a Hessenberg matrix, we begin with the recent unitary similarity transform into a triangular plus rank-one matrix. Then we show nonunitary transforms of this matrix at a low arithmetic cost into similar arrow-head matrices. The resulting eigenproblem can be effectively solved by the known algorithms. Based on some properties of the TPRI matrices, we also show that the similarity transforms into both Hessenberg and TPRI forms tend to decrease the geometric multiplicities of the eigenvalues, and we discuss some relevant research topics.