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 $$Av=\lambda v$$.