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# Math Seminar: "Constructive Proof of Hessenberg Form"

2015-09-28

4:10pm Neill 5W

Thomas Cameron

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.