Eigenspace

Eigenspace vs. matrix subspaces

Unlike the column, row, and null spaces, the eigenspace only exists for a square matrix. The eigenspace of a square matrix, $A$, is the set of vectors that preserve their direction under the linear transformation matrix, $A$. Furthermore, eigenspace can be complex in contrast to the other spaces of a matrix, which are always real.

Definition

Formally, an eigenspace of an $n\times n$ matrix, $A$, is defined as:

$$E_\lambda(A)=\{\bold{x}|A\bold{x=\lambda x}\}$$

, where $\lambda$ is an eigenvalue of $A$ and $\bold{x}$ is the corresponding eigenvector of $A$. Thus, eigenspace consists of all the eigenvectors of a matrix corresponding to an eigenvalue.

Example

Consider a matrix, $A=\begin{bmatrix}3 & 2\\3 & 4\end{bmatrix}$. Furthermore, consider an eigenvalue, $\lambda=6$, and the corresponding eigenvector, $\bold{x}=\begin{bmatrix}10\\15\end{bmatrix}$. Notice that all multiples of $\bold{x}$ (that is $span(\bold{x})$) would also make valid eigenvectors corresponding to the same eigenvalue, such as $\frac{1}{5}\bold{x}=\begin{bmatrix}2 \\ 3\end{bmatrix}$. Therefore, by definition, the eigenspace corresponding to eigenvalue 6 will have the span of $\bold{x}$.

Note: There are infinitely many eigenvectors corresponding to each eigenvalue of a matrix.

Below is a visualization that shows how eigenvectors of a matrix preserve their direction under the linear transformation of that matrix. The red and blue vectors are eigenvectors corresponding to the eigenvalue, $6$. These vectors kept their direction under the matrix A transformation. In contrast, a brown vector loses its ...