Eigenvectors and Eigenvalues

Eigenvectors and eigenvalues. Let

be an $n\times n$ square matrix, and let

be the $n\times n$ identity matrix. Suppose that $\lambda$ be a scalar, and that the matrix equation

$\displaystyle A \mathbf{v} = \lambda\mathbf{v}$

has nontrivial solutions. Then the scalar $\lambda$ is called an eigenvalue of

, and a nontrivial solution $\mathbf{v}$ is an eigenvector associated with $\lambda$ .

Diagonal matrix. Let $\lambda_1,\ldots,\lambda_n$ be scalars. Then the $n\times n$ square matrix

$\displaystyle D = \begin{bmatrix} \lambda_1 & 0 & \cdots & 0 \\ 0 & \lambda_2 & \cdots & 0 \\ \hdotsfor{4} \\ 0 & 0 & \cdots & \lambda_n \end{bmatrix}$

is called the diagonal matrix with diagonal entries $\lambda_1,\ldots,\lambda_n$ .

Matlab/Octave. The function diag(v) with column vector v returns the diagonal matrix with diagonal entries given by the vector v. The function eig(A) returns a vector containing eigenvalues of . The command

[V,D] = eig(A)

returns a diagonal matrix

with eigenvalues on diagonal entries, and a matrix

whose column vectors are the corresponding eigenvectors.

EXAMPLE 1. Each of the matrices below answer the following questions: (a) Find the eigenvalue corresponding to the eigenvector $\mathbf{v}$ . (b) Corresponding to the eigenvalue in (a), find all the eigenvectors.

$A = \begin{bmatrix} 1 & 6 \\ 5 & 2 \end{bmatrix}$ with $\mathbf{v} = \begin{bmatrix}6 -5 \end{bmatrix}$
$A = \begin{bmatrix} 4 &-1 & 6 \\ 2 & 1 & 6 \\ 2 &-1 & 8 \end{bmatrix}$ with any vector $\mathbf{v} \in$ span $\left\{ \begin{bmatrix}1 2 0 \end{bmatrix}, \begin{bmatrix}-3 0 1 \end{bmatrix} \right\}$