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Washington State University

Combinatorics, Linear Algebra and Number Theory (CLaN) Seminar

Department of Mathematics

Neill Hall 5W

September 28, Monday, 4:10 - 5:00 PM

Thomas R. Cameron

Department of Mathematics

Washington State University

Combinatorics, Linear Algebra and Number Theory (CLaN) Seminar

Department of Mathematics

Neill Hall 5W

September 28, Monday, 4:10 - 5:00 PM

Thomas R. Cameron

Department of Mathematics

Washington State University

Title: Constructive Proof of Hessenberg Form

Abstract: Every real and complex matrix is unitarily similar to matrix in Hessenberg form. This similarity transformation can be done in a finite number of steps. Moreover, the QR and QZ algorithms are made cost efficient by a preliminary reduction to Hessenberg and Hessenberg-Triangular form, respectively.

A matrix polynomial, P(z), is a matrix whose entries are scalar polynomials with real or complex coefficients. In this talk we will present a construction of Hessenberg form for matrix polynomials. A preliminary reduction to Hessenberg form has the possibility of making several algorithms for computing the eigenvalues of a matrix polynomial cost efficient.

We will conclude with possible areas of future research, which include numerically stable algorithms for computing this Hessenberg form.

Abstract: Every real and complex matrix is unitarily similar to matrix in Hessenberg form. This similarity transformation can be done in a finite number of steps. Moreover, the QR and QZ algorithms are made cost efficient by a preliminary reduction to Hessenberg and Hessenberg-Triangular form, respectively.

A matrix polynomial, P(z), is a matrix whose entries are scalar polynomials with real or complex coefficients. In this talk we will present a construction of Hessenberg form for matrix polynomials. A preliminary reduction to Hessenberg form has the possibility of making several algorithms for computing the eigenvalues of a matrix polynomial cost efficient.

We will conclude with possible areas of future research, which include numerically stable algorithms for computing this Hessenberg form.