Eigenvalues and Eigenvectors

≈ 25 min

For a square matrix $A$ , a non-zero vector $\mathbf{v}$ is an eigenvector with eigenvalue $\lambda$ if

A\mathbf{v} = \lambda \mathbf{v}.

The matrix acts on such a vector by pure scaling — no change of direction. Eigenvalues are the roots of the characteristic equation

\det(A - \lambda I) = 0.

Worked example. For $A = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$ , $\det(A - \lambda I) = (2-\lambda)^2 - 1 = \lambda^2 - 4\lambda + 3 = (\lambda - 1)(\lambda - 3).$ So $\lambda = 1$ or $\lambda = 3$ . For $\lambda = 3$ , solving $(A - 3I)\mathbf{v} = 0$ gives $\mathbf{v} = (1, 1)$ ; for $\lambda = 1$ , $\mathbf{v} = (1, -1).$

What are the eigenvalues of the diagonal matrix $\begin{pmatrix} 5 & 0 \\ 0 & -2 \end{pmatrix}$ ?

$5$ and $-2$

$0$ and $0$

$5$ and $2$

$-2$ and $-5$

Find the largest eigenvalue of $A = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$ .

The product of the eigenvalues of a matrix equals its determinant. Find that product for $A = \begin{pmatrix} 4 & 2 \\ 1 & 3 \end{pmatrix}$ .

Which vector is an eigenvector of $A = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$ for the eigenvalue $\lambda = 3$ ?

$(1, 1)$

$(1, -1)$

$(1, 0)$

$(0, 1)$