HessenbergForm[F](A) returns the upper Hessenberg form H of A.

Given an n x n Matrix A of elements in a field F, the algorithm converts a copy of A into upper Hessenberg form H using O(n^3) operations in F. The algorithm requires that F be a field and should only be used if F is finite as there is severe expression swell in computing H.

The (indexed) parameter F, which specifies the domain of computation, a field, must be a Maple table/module which has the following values/exports:

F[`0`]: a constant for the zero of the ring F

F[`1`]: a constant for the (multiplicative) identity of F

F[`+`]: a procedure for adding elements of F (nary)

F[`-`]: a procedure for negating and subtracting elements of F (unary and binary)

F[`*`]: a procedure for multiplying two elements of F (commutative)

F[`/`]: a procedure for dividing two elements of F

F[`=`]: a boolean procedure for testing if two elements in F are equal

$\mathrm{with}\left(\mathrm{LinearAlgebra}\left[\mathrm{Generic}\right]\right)\:$

$Q\left[\mathrm{`0`}\right],Q\left[\mathrm{`1`}\right],Q\left[\mathrm{`+`}\right],Q\left[\mathrm{`-`}\right],Q\left[\mathrm{`*`}\right],Q\left[\mathrm{`/`}\right],Q\left[\mathrm{`=`}\right]≔0,1,\mathrm{`+`},\mathrm{`-`},\mathrm{`*`},\mathrm{`/`},\mathrm{`=`}\:$

$A≔\mathrm{Matrix}\left(\left[\left[2\,-7\,-3\,4\right]\,\left[1\,-3\,-4\,5\right]\,\left[-7\,10\,5\,-7\right]\,\left[-7\,10\,5\,-7\right]\right]\right)$

$A≔\left[\begin{array}{cccc}2& \mathrm{-7}& \mathrm{-3}& 4\\ 1& \mathrm{-3}& \mathrm{-4}& 5\\ \mathrm{-7}& 10& 5& \mathrm{-7}\\ \mathrm{-7}& 10& 5& \mathrm{-7}\end{array}\right]$

$H≔\mathrm{HessenbergForm}\left[Q\right]\left(A\right)$

$H≔\left[\begin{array}{cccc}2& \mathrm{-14}& 1& 4\\ 1& \mathrm{-10}& 1& 5\\ 0& \mathrm{-46}& 5& 28\\ 0& 0& 0& 0\end{array}\right]$

$H,U≔\mathrm{HessenbergForm}\left[Q\right]\left(A\,\mathrm{output}=\left['H'\,'U'\right]\right)$

$H,U≔\left[\begin{array}{cccc}2& \mathrm{-14}& 1& 4\\ 1& \mathrm{-10}& 1& 5\\ 0& \mathrm{-46}& 5& 28\\ 0& 0& 0& 0\end{array}\right],\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 7& 1& 0\\ 0& 0& \mathrm{-1}& 1\end{array}\right]$

$\mathrm{MatrixMatrixMultiply}\left[Q\right]\left(\mathrm{MatrixMatrixMultiply}\left[Q\right]\left(U\,A\right)\,\mathrm{MatrixInverse}\left[Q\right]\left(U\right)\right)$

$\left[\begin{array}{cccc}2& \mathrm{-14}& 1& 4\\ 1& \mathrm{-10}& 1& 5\\ 0& \mathrm{-46}& 5& 28\\ 0& 0& 0& 0\end{array}\right]$