The Hessenberg indexing function can be used to construct rtable objects of type Array or Matrix.

In the construction of a Matrix, if Hessenberg or Hessenberg[upper] is included in the calling sequence as an indexing function (shape), an upper Hessenberg Matrix is returned.

Note:  A Hessenberg Matrix is a triangular Matrix with one extra (contiguous) diagonal.

This indexing function may also be qualified as Hessenberg[lower].

The specification is similar in the construction of an Array.

If an object is defined by using the Hessenberg or Hessenberg[upper] indexing function, the elements located in the lower triangle, below the subdiagonal, cannot be reassigned.

The situation is similar in a construction that uses Hessenberg[lower] as an indexing function.

$A≔\mathrm{Array}\left(\mathrm{Hessenberg}\left[\mathrm{upper}\right]\,1..4\,1..4\,\left(a\,b\right)\mapsto a+b\right)$

$A≔\left[\begin{array}{cccc}2& 3& 4& 5\\ 3& 4& 5& 6\\ 0& 5& 6& 7\\ 0& 0& 7& 8\end{array}\right]$

$M≔\mathrm{Matrix}\left(3\,\left[\left[1\,1\,1\right]\,\left[1\,1\,1\right]\,\left[1\,1\,1\right]\right]\,\mathrm{shape}=\mathrm{Hessenberg}\left[\mathrm{lower}\right]\right)$

$M≔\left[\begin{array}{ccc}1& 1& 0\\ 1& 1& 1\\ 1& 1& 1\end{array}\right]$

$M\left[1,2\right]≔5$

$M_{1,2}≔5$

$M$

$\left[\begin{array}{ccc}1& 5& 0\\ 1& 1& 1\\ 1& 1& 1\end{array}\right]$

$M\left[1,3\right]≔5$

Error, attempt to assign to upper triangle of lower Hessenberg rtable