TITLE: What is a Dominant Eigenvalue?
QUESTION [8 upvotes]: I know this is a rather simple questions. I've searched for a while and from what i gather about the dominant Eiganvalue.
If i have $\lambda_1$ = 1 $\lambda_2$ = 3 and $\lambda_3$=5 (hypothetically) would 5 be the dominant Eiganvalue? And from that can i assume that given any set of eigenvalues the one with highest weight is always the Dominant?

REPLY [2 votes]: If the eigenvalues of an $n \times n$ matrix $A$ can be arranged as
$$
| \lambda_1 | > | \lambda_2 | \geq | \lambda_3 | \geq \cdots \geq | \lambda_n |
$$
then $\lambda_1$ is called the dominant eigenvalue of $A$.
The dominant eigenpair $(\lambda_1, \mathbf{v}_1)$ of $A$ is very useful for determining the steady-state (long-term behavior) of linear dynamical systems of the form
$$
\dot{\mathbf{x}} = A \mathbf{x}
$$
or
$$
\mathbf{x}_{n + 1} = A \mathbf{x}_n
$$
Power method gives a simple numerical procedure for finding the dominant eigenvalue $\lambda_1$ of $A$ (when it exists) and the Inverse Power Method gives a simple numerical procedure for finding the corresponding (dominant) eigenvector of $A$.