Hessenberg reduction

Performs Householder transformations since a Hessenberg matrix is obtained. A Hessenberg matrix is a matrix of the form

$\left(\begin{array}{cccccc}{a}_{11}& a_{12}& a_{13}& \cdots & a_{1(n-1)}& a_{1n}\\ a_{21}& a_{22}& a_{23}& \cdots & a_{2(n-1)}& a_{2n}\\ 0& a_{32}& a_{33}& \cdots & a_{3(n-1)}& a_{3n}\\ 0& 0& a_{43}& \cdots & a_{4(n-1)}& a_{4n}\\ 0& 0& 0& \ddots & a_{5(n-1)}& a_{5n}\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ 0& 0& 0& a_{(n-1)(n-2)}& a_{(n-1)(n-1)}& a_{(n-1)n}\\ 0& 0& 0& 0& a_{n(n-1)}& a_{nn}\end{array}\right)$

hessenberg_reduction(Matrix))

Description

Given a matrix, transforms it to a Hessenberg matrix.