A (upper) Hessenberg matrix (according to Karl Hessenberg ) is a square matrix whose entries are equal to zero below the first subdiagonal, ie for all.
Analogously one defines a lower Hessenberg matrix as a square matrix whose transpose is an upper Hessenberg matrix. If only by a Hessenberg matrix, the speech, an upper Hessenberg matrix is mostly meant.
A matrix which is both a lower and an upper mountain Hesse matrix are called tridiagonal.
Hessenberg matrices occur naturally in Krylov subspace methods, and as a preliminary step in the computation of eigenvalues using the QR algorithm. The numerical transformation of an arbitrary 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.
- Numerical linear algebra