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.

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

The vector space supplies both the vectors and the scalar multiplication in $\boldsymbol{A}\boldsymbol{p}=\lambda\boldsymbol{p}$$\lambda \mathbf{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}$$\left(\begin{array}{cc}2& 0\\ 0& 3\end{array}\right)$, $(1,0)$ is an eigenvector with eigenvalue $2$.
• For the same matrix, $(0,1)$ is an eigenvector with eigenvalue $3$.

Prerequisites¶

• Linear Operator
• Vector Space
• Field

Related Concepts¶

• Characteristic Polynomial
• Jordan Normal Form
• Zero Vector
• Invariant Subspace
• Matrix

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

Last updated: 20 October 2026 03:02