Show that all upper triangular matrices with diagonal entries and arbitrary other elements are equivalent if are nonzero
Consider
with , then
is the only eigenvalue of (multiplicity ).
Besides,
.
This means that the canonical form of Jordan for has only one block:
As a consequence, all matrices are equivalent to (even more, similar to ).
Regards.
Fernando Revilla
I add the following to my previous post:
(i) If then, and all matrices are equivalent to (by a well known theorem).
(ii) If then, and all matrices are equivalent to
(by a well known theorem).
So, we can avoid similarity and only use the concept of equivalence.
Regards.
Fernando Revilla