Eigenvectors and Eigenvalues

Eigenvector and Eigenvalue

Definition

Given an $\left( n \cdot m \right)$ matrix $A$ and an $n$-vector $\overrightarrow{r}$, if $\overrightarrow{r^{\prime}} = Ar$ points in the same direction as $\overrightarrow{r}$, i.e. $\overrightarrow{r^{\prime}} = \lambda \overrightarrow{r}$ where $\lambda$ is a real scalar, then $r$ is called an eigenvector of $A$ with real eigenvalue.
The cases $r = 0$ or $\lambda = 0$ are excluded from this definition.

Trivial Eigenvalues

Theorem

If the matrix equation $A \overrightarrow{v} = \lambda \overrightarrow{v}$, is inversible we have the trivial solutions:

$$ A = \begin{pmatrix} \lambda &0\\ 0&\lambda \end{pmatrix} $$

and

$$ \overrightarrow{v} = \overrightarrow{0}$$

Proof

$$ A \overrightarrow{v} = \lambda \overrightarrow{v} $$ $$ \Leftrightarrow \left( A - \lambda \mathbb{1} \right) \overrightarrow{v} = \overrightarrow{0}$$

If $A$ is reversible:

$$ \Leftrightarrow A = \lambda \mathbb{1} \vee \overrightarrow{v} = (A- \lambda \mathbb{1})^{-1}\ \overrightarrow{0} $$ $$ \Leftrightarrow A = \begin{pmatrix} \lambda &0\\ 0&\lambda \end{pmatrix} \vee \overrightarrow{v} = \overrightarrow{0} $$

$\square$

Characteristic equation

When solving for non trivial eigenvalues we call the following equation the characteristic equation:

$$ \det \left( A - \lambda \mathbb{1} \right) = 0$$

Characteristic polynomial

We call $A - \lambda \mathbb{1}$, the characteristic polynomial of A, when solving for eigenvalues.

Non-trivial Eigenvalues

Theorem 1

If $(A - \lambda \mathbb{1})$ does not have an inverse, $\overrightarrow{v} \neq 0$.

Theorem 2

For the real scalar $\lambda$ to be an eigenvalue of the matrix $A$ it must be a real root of the characteristic equation:

$$ \det \left( A - \lambda \mathbb{1} \right) = 0 $$

Remark

This is a polynomial equation of degree $n$ if $A$ is an $\left( n \cdot n \right)$ matrix.

Multiple of eigenvector solutions

Theorem

For any solution of eigenvectors $\overrightarrow{v}$ there is a multiple of the eigenvector $\alpha \in \mathbb{R}$ that still satisfies the equation:

$$ A \alpha \overrightarrow{v} = \lambda \alpha \overrightarrow{v} $$