Hessenberg matrix

A Hessenberg matrix is a special class of square matrices that is particularly considered in the mathematical sub-area of numerical linear algebra . These matrices are named after Karl Hessenberg .

definition

An (upper) Hessenberg matrix is ​​a square matrix whose entries below the first secondary diagonal are equal to zero, i.e. for all . ${\displaystyle H\in \mathbb {C} ^{n\times n}}$ ${\displaystyle h_{ij}=0}$ ${\displaystyle i>j+1}$

${\displaystyle H={\begin{pmatrix}h_{11}&h_{12}&h_{13}&\cdots &h_{1n}\\h_{21}&h_{22}&h_{23}&\cdots &h_{2n}\\0&h_{32}&h_{33}&\cdots &h_{3n}\\\vdots &\ddots &\ddots &\ddots &\vdots \\0&\cdots &0&h_{nn-1}&h_{nn}\end{pmatrix}}}$

Similarly, the lower Hessenberg matrix is ​​defined as a square matrix whose transpose is an upper Hessenberg matrix. If only a Hessenberg matrix is ​​mentioned, an upper Hessenberg matrix is ​​usually meant.

A matrix that is both a lower and an upper Hessenberg matrix is ​​a tridiagonal matrix .

application

Hessenberg matrices occur naturally in the Krylow subspace method and as a preliminary stage in the calculation of eigenvalues ​​using the QR algorithm . The numerical transformation of any matrix to Hessenberg form is described in the QR algorithm. The structure of the matrices is reflected in the inverse , the adjuncts and the eigenvectors .

Individual evidence

1. ^ Hessenberg form . In: Guido Walz (Ed.): Lexicon of Mathematics . 1st edition. Spectrum Academic Publishing House, Mannheim / Heidelberg 2000, ISBN 3-8274-0439-8 .