Editing Hessenberg matrix (section)

{{Short description|Kind of square matrix in linear algebra}}
In [[linear algebra]], a '''Hessenberg matrix''' is a special kind of [[square matrix]], one that is "almost" [[Triangular matrix|triangular]]. To be exact, an '''upper Hessenberg matrix''' has zero entries below the first [[diagonal#Matrices|subdiagonal]], and a '''lower Hessenberg matrix''' has zero entries above the first [[Diagonal#Matrices|superdiagonal]].<ref>{{harvtxt|Horn|Johnson|1985}}, page 28; {{harvtxt|Stoer|Bulirsch|2002}}, page 251</ref> They are named after [[Karl Hessenberg]].<ref>Biswa Nath Datta (2010) Numerical Linear Algebra and Applications, 2nd Ed., Society for Industrial and Applied Mathematics (SIAM) {{ISBN|978-0-89871-685-6}}, p. 307</ref>

A '''Hessenberg decomposition''' is a [[matrix decomposition]] of a matrix <math>A</math> into a [[unitary matrix]] <math>P</math> and a Hessenberg matrix <math>H</math> such that <math display=block>PHP^*=A</math> where <math>P^*</math> denotes the [[conjugate transpose]].