Eigenbasis / Eigenspace

Given a eigenvalue $\lambda$ of matrix $A$, the null space (kernel) of $A-\lambda I$ is called the eigenspace of $A$ associated with $\lambda$.

In another way, eigenspace $E_{\lambda ;A}=\{\mathbf{v}\mid (A-\lambda I)\mathbf{v}=\mathbf{0}\}$ is the set of all eigenvectors associated with $\lambda$ plus zero vector.

If all the eigenvectors of $A_{n\times n}$ can form a basis of ${\mathbb{R}}^n$, they are called an eigenbasis of ${\mathbb{R}}^n$.

• So eigenbasis is not a basis of eigenspace.