In general if you convert to Hessenberg form a lot of what you might be trying to do with the matrix becomes easier to calculate. The QR algorithm was a common

In general if you convert to Hessenberg form a lot of what you might be trying to do with the matrix becomes easier to calculate. The QR algorithm was a common example of this. If you mean why would I want to put so much into simplifying linear algebra computations then you're missing out on the chance to work with some of the most useful stuff in comp sci (in my opinion)!