$\newcommand{\ones}{\mathbf 1}$

For any square matrix $A$, there is a invertible matrix $T$ for which $T^{-1}AT$ has Jordan form.

The Jordan form of a matrix can be determined from its characteristic polynomial.

Suppose $A$ has an independent set of $k$ eigenvectors.

The Jordan form of $A^2$ is the same as the Jordan form of $A$, except that each eigenvalue is squared.

If $A \in {\mathbf R}^{n \times n}$ is invertible, then its inverse can be expressed as $p(A)$, where $p$ is a polynomial with degree less than $n$.

The Jordan canonical form is frequently used in numerical computations.

Suppose $A \in \mathbf R^{3 \times 3}$. Then $A^3$ is a linear combination of $I$, $A$, and $A^2$.

The Jordan canonical form of the matrix $A$ is $J = \left[ \begin{array}{ccc} J_1 & & \\ & J_2 & \\ & & J_3 \end{array} \right]$ where the Jordan blocks $J_1$, $J_2$, and $J_3$ have sizes $1 \times 1$, $2 \times 2$, and $3 \times 3$, respectively, all associated with the same eigenvalue $\lambda$.

What is the rank of $\lambda I-A$?

What is the characteristic polynomial of $A$?

$A$ has an independent set of two eigenvectors.

$A$ has an independent set of four eigenvectors.