Eigenbasis¶

Definition¶

An eigenbasis is a basis of a vector space consisting entirely of eigenvectors of a linear operator.

Interpretation¶

An eigenbasis is a choice of coordinate directions in which the operator does nothing but stretch: each basis vector is sent to a multiple of itself. In an eigenbasis, the matrix of the operator is therefore diagonal, with the eigenvalues on the diagonal.

A square grid along the eigenvectors p1 and p2 mapped by A onto a grid stretched three times in the p1 direction and unchanged in the p2 direction, with v = p1 + p2 going to 3p1 + p2.

The left grid is drawn along the eigenvectors $\boldsymbol{p}_1$ and $\boldsymbol{p}_2$ of $\boldsymbol{A}=\begin{pmatrix}2&1\\1&2\end{pmatrix}$ $(2112)$ , with eigenvalues $3$ and $1$ . After applying $\boldsymbol{A}$ , the grid is stretched three times in the $\boldsymbol{p}_1$ direction and unchanged in the $\boldsymbol{p}_2$ direction. The vector $\boldsymbol{v}=\boldsymbol{p}_1+\boldsymbol{p}_2$ has coordinates $(1,1)$ in this basis and is sent to $3\boldsymbol{p}_1+\boldsymbol{p}_2$ $p_{2}$ , with coordinates $(3,1)$ ; in this basis the matrix of $\boldsymbol{A}$ is $\operatorname{diag}(3,1)$ .

In the eigenbasis $\boldsymbol{p}_1,\boldsymbol{p}_2$ the map only scales each basis vector, so its matrix is diagonal.

In general, if $\boldsymbol{v}=c_1\boldsymbol{p}_1+\cdots+c_n\boldsymbol{p}_n$ $c_{n} p_{n}$ , then $\boldsymbol{A}\boldsymbol{v}=\lambda_1c_1\boldsymbol{p}_1+\cdots+\lambda_nc_n\boldsymbol{p}_n$ $λ_{n} c_{n} p_{n}$ : each coordinate is multiplied by its own eigenvalue. Linear independence rules out several eigenvectors on one line, which cannot supply distinct coordinate directions, and the eigenvectors must span the whole space, so that every vector has such coordinates.

An eigenbasis exists exactly when the matrix is diagonalizable, and its vectors are the columns of $\boldsymbol{P}$ in diagonalization. The distinct eigenvalues theorem guarantees one when an $n\times n$ matrix has $n$ distinct eigenvalues. Not every matrix has one: the eigenvectors of the shear $\begin{pmatrix}1&1\\0&1\end{pmatrix}$ all lie on the horizontal axis.

An eigenbasis is not unique. Each eigenvector may be rescaled and the order may change, and for $\boldsymbol{I}_n$ every basis is an eigenbasis.

Key Results¶

A matrix is diagonalizable exactly when an eigenbasis exists.
The change-of-basis matrix $\boldsymbol{P}$ in diagonalization has eigenbasis vectors as its columns.

Examples¶

The standard basis is an eigenbasis for a diagonal matrix.
If two independent eigenvectors span $\mathbb{R}^2$ , they form an eigenbasis.

Prerequisites¶

Distinct Eigenvalues Theorem

Sources¶

Math II: Linear Algebra · Week 6

Aliases: eigenvector basis, basis of eigenvectors Symbols: lambda, real numbers

Last updated: 20 October 2026 03:02