Hessenberg SimilarityOne of the most important condensed forms in numerical linear algebra is theHessenberg matrix. An upper Hessenberg matrix is almost upper triangular, havingzeros below the first subdiagonal.Every square matrix is orthogonally similar to a Hessenberg matrix. There aremethods for reducingA∈Rn×ntoH=VtAVbased on Gram-Schmidt (theArnoldimethod) and on Householder reflectors (a slight modification to the HouseholderQRmethod). These methods are about twice as expensive as theirQRanalogs (theHouseholder Hessenberg reduction requires about103n3flops), but the resultingmatrix issimilartoA.There are several reasons Hessenberg matrices are important, but I would suggestthat the following is the most fundamental: Suppose that you wanted to solve(A-sI)x=bformdifferent values ofs. Gaussian elimination (orQR) wouldrequire a new factorization for eachs, resulting in a cost of O(mn3) flops. But(A-sI)x=b⇒Vt(A-sI)V Vtx=Vtb⇒(H-sI)y=z,
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Triangular matrix, Eigenvalue algorithm, QR factorization QR, Householder QR method