Hessenberg matrix

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 ${\displaystyle A}$ into a unitary matrix ${\displaystyle P}$ and a Hessenberg matrix ${\displaystyle H}$ such that $${\displaystyle PHP^{*}=A}$$ where ${\displaystyle P^{*}}$ denotes the conjugate transpose.