Eigenvalue and Eigenvector¶

Definition¶

For a linear operator or square matrix $\boldsymbol{A}$ , a nonzero vector $\boldsymbol{p}$ and scalar $\lambda$ satisfying $\boldsymbol{A}\boldsymbol{p}=\lambda \boldsymbol{p}$ give an eigenvector $\boldsymbol{p}$ and corresponding eigenvalue $\lambda$ .

Interpretation¶

For a real eigenvector with a real eigenvalue, the line it spans is invariant under the transformation. A positive eigenvalue preserves its direction, a negative eigenvalue reverses it, and a zero eigenvalue maps it to the zero vector. The equation $\boldsymbol{A}\boldsymbol{p}=\lambda \boldsymbol{p}$ also defines eigenvectors and eigenvalues over the complex numbers.

Two eigenvector directions of a matrix: p1 is stretched by 3 and p2 is kept, while another vector v is turned by the matrix.

$\boldsymbol{A}\boldsymbol{p}=\lambda \boldsymbol{p}$ with $\boldsymbol{p}\neq\boldsymbol{0}$ $0$ : $\boldsymbol{A}$ only rescales $\boldsymbol{p}$ , by the eigenvalue $\lambda$ . Here $\boldsymbol{A}=\begin{pmatrix}2&1\\1&2\end{pmatrix}$ has $\lambda_1=3$ on $\boldsymbol{p}_1=\binom{1}{1}$ and $\lambda_2=1$ on $\boldsymbol{p}_2=\binom{1}{-1}$ $(- 1 1)$ ; a vector $\boldsymbol{v}$ off these lines is turned.

Turn the unit vector $\boldsymbol{x}$ with $\varphi$ : usually $\boldsymbol{A}\boldsymbol{x}$ points elsewhere, but on the dashed eigenvector lines $\boldsymbol{A}\boldsymbol{x}=\lambda \boldsymbol{x}$ $λ x$ . A quarter turn has no real eigenvector.

The vector space supplies both the vectors and the scalar multiplication in $\boldsymbol{A}\boldsymbol{p}=\lambda\boldsymbol{p}$ $λ p$ ; this equation compares two vectors in that same space.

Key Results¶

The zero vector is excluded from being an eigenvector.
Eigenvalues of triangular or diagonal matrices are their diagonal entries.
For a real-valued matrix, nonreal complex eigenvalues and eigenvectors occur in complex conjugate pairs.

Examples¶

For $\boldsymbol{A}=\begin{pmatrix}2&0\\0&3\end{pmatrix}$ $(2003)$ , $(1,0)$ is an eigenvector with eigenvalue $2$ .
For the same matrix, $(0,1)$ is an eigenvector with eigenvalue $3$ .

Prerequisites¶

Sources¶

Math II: Linear Algebra · Week 6
Math V: Advanced Linear Algebra · Week 1 (revisited)

Aliases: characteristic root, latent root, proper value, characteristic value, characteristic vector, latent vector Keywords: eigenvalue, eigenvector, eigenvalues, eigenvectors, proper vector, eigenpair Symbols: lambda, phi

Last updated: 20 October 2026 03:02