Alright, doesn't surprise me. Thanks for the help.
Indeed it shouldn't surprise you. There are no square-root of the matrix
, for example. The theory doesn't want non-invertibles for a reason.
On the other hand, diagonalisability is something we can get around, at least in finite dimensions, by using the Jordan normal form (infinite-dimensional Jordan normal form is highly sensitive to small perturbations, plus other technical problems like the presence of pseudospectrum, which makes them rather useless in general). For example,
has a square-root, namely
, which you can obtain from the series
, N nilpotent.