Eigenbasis

linear-algebra

Definition

Eigenbasis

Let $T:V\to V$ be a linear operator on a finite-dimensional vector space $V$. An eigenbasis of $T$ is a basis of $V$ consisting entirely of eigenvectors of $T$.

Formally, a basis $B=(v_1,\dots ,v_n)$ is an eigenbasis if, for each $i$, there exists an eigenvalue ${\lambda }_i$ such that

$$T{v}_i={\lambda }_i{v}_i.$$

In an eigenbasis, the operator acts by independent scaling of coordinates:

$$T\left(\sum _{i=1}^n{c}_i{v}_i\right)=\sum _{i=1}^n{c}_i{\lambda }_i{v}_i.$$

Thus, the matrix of $T$ in this basis is diagonal, with the eigenvalues on the diagonal.