Ranks of shifted powers determine Jordan form up to block order

Fix an eigenvalue $\lambda$. On $G_{\lambda }$, $T-\lambda I=N_{\lambda }$. On $G_{\mu }$ with $\mu \mathrm{\ne }\lambda$, it is $(\mu -\lambda )I+N_{\mu }$, which is invertible because the finite geometric series in $-N_{\mu }\mathrm{/}(\mu -\lambda )$ is an inverse up to the nonzero scalar $\mu -\lambda$.

By the direct sum in [L1] and uniqueness in [L4], for every $k\ge 0$ the rank on the non-$\lambda$ summands is their constant total dimension $c_{\lambda }$, while the rank on $G_{\lambda }$ is $\mathrm{rank}{N}_{\lambda }^k$; hence ${\rho }_k(\lambda )=c_{\lambda }+\mathrm{rank}{N}_{\lambda }^k$.

The constant $c_{\lambda }$ cancels from the second difference, and [L2] then gives the displayed exact-size block count. Varying $k$ and $\lambda$ recovers the entire block multiset.

By the definition of Jordan form, a block multiset determines the block diagonal matrix up to block order; for $V=0$ there are no eigenvalues or blocks.