Hessenberg - Maple Help

The Hessenberg Indexing Function

Description

 • 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.

Examples

 > $A≔\mathrm{Array}\left(\mathrm{Hessenberg}\left[\mathrm{upper}\right],1..4,1..4,\left(a,b\right)↦a+b\right)$
 ${A}{≔}\left[\begin{array}{cccc}{2}& {3}& {4}& {5}\\ {3}& {4}& {5}& {6}\\ {0}& {5}& {6}& {7}\\ {0}& {0}& {7}& {8}\end{array}\right]$ (1)
 > $M≔\mathrm{Matrix}\left(3,\left[\left[1,1,1\right],\left[1,1,1\right],\left[1,1,1\right]\right],\mathrm{shape}=\mathrm{Hessenberg}\left[\mathrm{lower}\right]\right)$
 ${M}{≔}\left[\begin{array}{ccc}{1}& {1}& {0}\\ {1}& {1}& {1}\\ {1}& {1}& {1}\end{array}\right]$ (2)
 > $M\left[1,2\right]≔5$
 ${{M}}_{{1}{,}{2}}{≔}{5}$ (3)
 > $M$
 $\left[\begin{array}{ccc}{1}& {5}& {0}\\ {1}& {1}& {1}\\ {1}& {1}& {1}\end{array}\right]$ (4)
 > $M\left[1,3\right]≔5$