I have come across the following canonical form, which is unfamiliar to me:
Let $A,B\in\mathbb C^{n\times n}$ and suppose there exists $P\in \mathbb C^{n\times n}$ such that $$A=PBP^{-1}=\begin{pmatrix}J&0\\0&N\\ \end{pmatrix},$$ where $J\in\mathbb C^{k\times k}$ invertible and $N\in\mathbb C^{(n-k)\times (n-k)}$ nilpotent.
Here, both $J,N$ are in Jordan canonical form (JCF). What confuses me is that this canonical form above is also referred to as the Jordan canonical form. So far, I have never seen (or perhaps, just not yet appreciated) that the JCF 'separates' a square complex matrix into a nonsingular part and a nilpotent part.
How does one derive the above form from the usual JCF? Or, are they in fact two distinct forms (with the naming of this as also the JCF being a mistake)? Are there any examples to elucidate the above form?
The nilpotent sub-matrix $N$ consists of all the Jordan blocks corresponding to eigenvalue $0$, while the other sub-matrix $J$ consists of the Jordan blocks corresponding to nonzero eigenvalues. If arrange the "usual JCF" so that the zero-eigenvalue blocks are together on the lower right, then you have the above decomposition.