[mamonster]

I'm open to being corrected, but AFAIK the normal form (1870) precedes the official focus on operators (with Hilbert) by like 20-30 years.

[srean]

KalMann is correct. Jordan canonical form decomposition is more general. Every matrix in an algebraically closed field will have such a decomposition. This is not true for spectral decomposition. Only diagonalizable matrices will have a spectral decomposition and they are a smaller subset.

That said, Jordan form is uglier than spectral decomposition, to my taste that is. Spectral decomposition so beautiful and neat.