Up to now, we've been reviewing the structure of vector spaces and linear transformations. Now we start on the heart of the course: eigenvalues and eigenvectors. As with all mathematical topics, you should ask yourself three questions:

1. What is it?
2. How do you compute it?
3. What is it good for?

If $A$ is an $n \times n$ matrix, and if ${\bf x}$ is a non-zero vector such that $$A{\bf x} = \lambda {\bf x}$$ for some scalar $\lambda$, then we say that ${\bf x}$ is an eigenvector with eigenvalue $\lambda$. For each eigenvalue $\lambda$, the set of all solutions to $A{\bf x} =\lambda {\bf x}$ is called an eigenspace and is denoted $E_\lambda$.

If $L: V \to V$ is an operator, the definitions are similar. If ${\bf v} \in V$ is a non-zero vector such that $L({\bf v}) = \lambda {\bf v}$, then ${\bf v}$ is an eigenvector with eigenvalue $\lambda$. The eigenvalues and eigenvectors of the operator $L$ are closely related to the eigenvalues and eigenvectors of the matrix $[L]_\mathcal{B}$, where $\mathcal{B}$ is a basis for $V$.