Page "Hessenberg matrix" Paragraph 6

Many linear algebra algorithms require significantly less computational effort when applied to 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's algorithm of unitary similarity transforms ).

Subsequent reduction of Hessenberg matrix to a triangular matrix can be achieved through iterative procedures, such as shifted QR-factorization.

In eigenvalue algorithms, 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.