Eigenvectors and Eigenvalues

By definition, $$\lambda \in \mathbb{C}$$ is an eigenvalue of $$A \in \mathbb{C}^{n\times n}$$ if

$$\mathcal{X}(\lambda) = \det(\lambda I - A)$$.

This is equivalent to the following two things:

1) There exists a (right) eigenvector: There exists non-zero $$v \in \mathbb{C}^n$$ such that $$(\lambda I -A) v = 0$$, i.e.,

$$Av = \lambda v$$ any such $$v$$ is called an eigenvector of $$A$$ associated with eigenvalue $$\lambda$$.

2) There exists a left eigenvector: There exists non-zero $$w \in \mathbb{C}^N$$ s.t. $$w^T(\lambda I - A) = 0$$, i.e.,

$$w^T A = \lambda w^T$$. Any such $$w$$

Properties of Eigenvectors and Eigenvalues

 * If $$v$$ is an eigenvecotr of A with eigenvalue $$\lambda$$, then so is $$av$$, for any $$\alpha in \mathbb{C}$$, $$a\neq 0$$
 * Even when $$A$$ is real, $$\lambda$$ and $$v$$ can be complex
 * When $$A$$ and $$\lambda$$ are real, we can always find a real eigenvector associated with $$\lambda$$: if $$Av = \lambda v$$ with $$A \in \mathbb{R}^{n\times n}$$, $$\lambda \in \mathbb{R}$$ and $$v \in \mathbb{C}^n$$, then $$A \Re v = \lambda \Re v$$ and $$A \Im v = \lambda \Im v$$. In other words, $$\Re v$$ and $$\Im v$$ are real eigenvectors, if they are non-zero (and at least one is).
 * Conjugate symmetry: if $$A$$ is real and $$v \in \mathbb{C}^n$$ is an eigenvector associated with $$\lambda \in \mathbb{C}$$, then $$\bar{v}$$ is an eigenvector associated with $$\bar{\lambda}$$. (Can be shown by taking conjugate of $$Av=\lambda v$$.

What's the point of Eigenvalues or Eigenvectors?
Eigenvalues or eigenvectors are ubiquitous, and any list that will be compiled will be largely incomplete. But here are some purposes and uses of eigenvalues/eigenvectors...


 * The conditions for the stability of a continuous-time LDS can be expressed via left or right eigenvectors.
 * The conditions for the stability of a discrete-time LDS can be expressed eigenvalues (see Modal Form).
 * Understanding behavior of LDSs: If the initial condition of an LDS is an eigenvector, then we know that the system trajectory will remain within the line spanned by that eigenvector

TBC