Preserving the Hessenberg shape in operations with upper triangular matrices would be desirable

[andreasvarga]

I found the support for the Hessenberg form of a matrix very useful. It would be also useful to have a similar support for the generalized Hessenberg form of a matrix pair. In this context, I have the following issue to raise to your attention.

Operations involving Hessenberg matrices and upper triangular matrices as addition and multiplications preserve the Hessenberg form. Thus if Ah is a Hessenberg matrix and At is an upper triangular matrix, then operations like αAh + βAt, At*Ah , Ah*At, At\Ah or Ah/At produce results which are Hessenberg matrices too. This feature is presently supported only for the addition with UniformScalings, i.e., Ah+βI results with Hessenberg shape. Surprisingly Ah*I does not result Hessenberg, although it trivially is.