Similar Matrices
Let $M$ be any invertible matrix. Then $B=M^{1}AM$ is similar to $A$. Similar matrices $A$ and $M^{1}AM$ have the same eigenvalues. If $x$ is an eigenvector of $A$, then $M^{1}x$ is an eigenvector of $B=M^{1}AM$. Matrices that are similar to each other form a family of related matrices, all having the same eigenvalues.
If $A$ has $s$ independent eigenvectors, it is similar to a matrix $J$ that has $s$ Jordan blocks on its diagonal: Some matrix $M$ puts $A$ into Jordan form:
Jordan form $M^{1}AM=\left[\begin{array}{ccccc}J_1&& && && && \\ &&.&& && &&\\ && &&.&& &&\\ && && && &&J_s&&\end{array}\right] = J$ Each block in $J$ has one eigenvalue $\lambda_i$, one eigenvector and 1's above the diagonal: Jordan block $J_i=\left[\begin{array}{ccccc}\lambda_i&&1&& && && \\ &&. &&. && &&\\ && &&.&& &&\\ && && && &&\lambda_i&&\end{array}\right] = J$ $A$ is similar to $B$ if they share the same Jordan form $J$  not otherwise.
