Editing Hessenberg matrix (section)

==Computer programming==
Many linear algebra [[algorithm]]s require significantly less [[computational complexity theory|computational effort]] when applied to [[triangular matrix|triangular matrices]], and this improvement often carries over to Hessenberg matrices as well. If the constraints of a linear algebra problem do not allow a general matrix to be conveniently reduced to a triangular one, reduction to Hessenberg form is often the next best thing. In fact, reduction of any matrix to a Hessenberg form can be achieved in a finite number of steps (for example, through [[Householder transformation|Householder's transformation]] of unitary similarity transforms). Subsequent reduction of Hessenberg matrix to a triangular matrix can be achieved through iterative procedures, such as shifted [[QR decomposition|QR]]-factorization. In [[eigenvalue algorithm]]s, the Hessenberg matrix can be further reduced to a triangular matrix through Shifted QR-factorization combined with deflation steps. Reducing a general matrix to a Hessenberg matrix and then reducing further to a triangular matrix, instead of directly reducing a general matrix to a triangular matrix, often economizes the arithmetic involved in the [[QR algorithm]] for eigenvalue problems.