Editing Hessenberg matrix (section)

==Properties==
For <math>n \in \{1, 2\} </math>, it is [[Vacuous truth|vacuously true]] that every <math> n \times n </math> matrix is both upper Hessenberg, and lower Hessenberg.<ref>[https://www.cs.cornell.edu/~bindel/class/cs6210-f16/lec/2016-10-21.pdf Lecture Notes. Notes for 2016-10-21] Cornell University</ref>

The product of a Hessenberg matrix with a triangular matrix is again Hessenberg. More precisely, if <math>A</math> is upper Hessenberg and <math>T</math> is upper triangular, then <math>AT</math> and <math>TA</math> are upper Hessenberg.

A matrix that is both upper Hessenberg and lower Hessenberg is a [[tridiagonal matrix]], of which the [[Jacobi operator|Jacobi matrix]] is an important example. This includes the symmetric or Hermitian Hessenberg matrices. A Hermitian matrix can be reduced to tri-diagonal real symmetric matrices.<ref>{{Cite web|title=Computational Routines (eigenvalues) in LAPACK | url=http://sites.science.oregonstate.edu/~landaur/nacphy/lapack/eigen.html | website=sites.science.oregonstate.edu | access-date=2020-05-24}}</ref>