Symmetrizing a Hessenberg matrix: Designs for VLSI parallel processor arrays

A symmetrizer of a nonsymmetric matrix A is the symmetric matrix X that satisfies the equation XA = A(t)X, where t indicates the transpose. A symmetrizer is useful in converting a nonsymmetric eigenvalue problem into a symmetric one which is relatively easy to solve and finds applications in stability problems in control theory and in the study of general matrices. Three designs based on VLSI parallel processor arrays are presented to compute a symmetrizer of a lower Hessenberg matrix. Their scope is discussed. The first one is the Leiserson systolic design while the remaining two, viz., the double pipe design and the fitted diagonal design are the derived versions of the first design with improved performance.