There are no theorems in this section, just a selection of examples meant to illustrate the range of possibilities for the eigenvalues and eigenvectors of a matrix. These examples can all be done by hand, though the computation of the characteristic polynomial would be very time-consuming and error-prone. It can also be difficult to factor an arbitrary polynomial, though if we were to suggest that most of our eigenvalues are going to be integers, then it can be easier to hunt for roots. These examples are meant to look similar to a concatenation of Example CPMS3, Example EMS3 and Example ESMS3. First, we will sneak in a pair of definitions so we can illustrate them throughout this sequence of examples.
Suppose that $A$ is a square matrix and $\lambda$ is an eigenvalue of $A$. Then the algebraic multiplicity of $\lambda$, $\algmult{A}{\lambda}$, is the highest power of $(x-\lambda)$ that divides the characteristic polynomial, $\charpoly{A}{x}$.
Since an eigenvalue $\lambda$ is a root of the characteristic polynomial, there is always a factor of $(x-\lambda)$, and the algebraic multiplicity is just the power of this factor in a factorization of $\charpoly{A}{x}$. So in particular, $\algmult{A}{\lambda}\geq 1$. Compare the definition of algebraic multiplicity with the next definition.
Suppose that $A$ is a square matrix and $\lambda$ is an eigenvalue of $A$. Then the geometric multiplicity of $\lambda$, $\geomult{A}{\lambda}$, is the dimension of the eigenspace $\eigenspace{A}{\lambda}$.
Every eigenvalue must have at least one eigenvector, so the associated eigenspace cannot be trivial, and so $\geomult{A}{\lambda}\geq 1$.