In linear algebra, a Hessenberg matrix is a special kind of square matrix, one that is "almost" triangular. To be exact, an upper Hessenberg matrix has zero entries below the first subdiagonal, and a lower Hessenberg matrix has zero entries above the first superdiagonal.[1] They are named after Karl Hessenberg.[2]
A Hessenberg decomposition is a matrix decomposition of a matrix into a unitary matrix and a Hessenberg matrix such that where denotes the conjugate transpose.