Nilpotent Matrices and Jordan Structure

Can the index of an $n\times n$ nilpotent matrix ever exceed $n$?

Determine all possible Jordan forms $\mathbf{N}$ for a $4\times 4$ nilpotent matrix.

Explain why the number of blocks of size $i\times i$ or larger in the Jordan form for a nilpotent matrix is given by $\operatorname {rk}\left(\mathbf{L}^{i-1}\right) - \operatorname {rk}\left(\mathbf{L}^i\right)$.

For a nilpotent matrix $\mathbf{L}$ of index $k$, let $M_i = R(\mathbf{L}^i) \cap N(\mathbf{L})$. Prove that $M_i \subseteq M_{i-1}$ for each $i=0,1,\ldots ,k$.

Prove that $R(\mathbf{L}^{k-1}) \cap N(\mathbf{L}) = R(\mathbf{L}^{k-1})$ for all nilpotent matrices $\mathbf{L}$ of index $k>1$. In other words, prove $M_{k-1} = R(\mathbf{L}^{k-1})$.

Let $\mathbf{L}$ be a nilpotent matrix of index $k>1$. Prove that if the columns of $\mathbf{B}$ are a basis for $R(\mathbf{L}^i)$ for $i \le k-1$, and if $\left\{ \overrightarrow {v}_1, \overrightarrow {v}_2, \ldots , \overrightarrow {v}_s\right\}$ is a basis for $N(\mathbf{L}\mathbf{B})$, then $\left\{ \mathbf{B}\overrightarrow {v}_1, \mathbf{B}\overrightarrow {v}_2, \ldots , \mathbf{B}\overrightarrow {v}_s\right\}$ is a basis for $M_i$.

Find $\mathbf{P}$ and $\mathbf{N}$ such that $\mathbf{P}^{-1}\mathbf{L}\mathbf{P} = \mathbf{N}$ is in Jordan form, where

Determine the Jordan form for the following $8\times 8$ nilpotent matrix.

Prove that if $\mathbf{N}$ is the Jordan form for a nilpotent matrix $\mathbf{L}$ as described below, then for any set of nonzero scalars $\left\{ \zeta_1, \zeta_2, \ldots , \zeta_t\right\}$, the matrix $\mathbf{L}$ is similar to a matrix $\tilde{\mathbf{N}}$ of the form

In other words, the 1's on the superdiagonal of the $\mathbf{N}_i$'s are artificial because any nonzero value can be forced onto the superdiagonal of any $\mathbf{N}_i$. What's important in the ``Jordan structure'' of $\mathbf{L}$ is the number and sizes of the nilpotent Jordan blocks (or chains) and not the values appearing on the superdiagonals of these blocks.

Jordan form for a nilpotent matrix: every nilpotent $\mathbf{L}_{n\times n}$ of index $k$ is similar to a block-diagonal matrix $\mathbf{P}^{-1}\mathbf{L}\mathbf{P} = \mathbf{N} = \operatorname {diag}(\mathbf{N}_1, \mathbf{N}_2, \ldots , \mathbf{N}_t)$, in which each $\mathbf{N}_j$ is a nilpotent matrix having ones on the superdiagonal and zeros elsewhere.