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