What is an Eigenspace?

Eigenspace is a vector subspace of $R^n$. For a matrix $A$ with $n \times n$ dimensions, $n$ eigenvalues are associated with it, and each eigenvalue $\lambda_i$ corresponds to an eigenvector $\vec{v}_i$.

This is known as the eigendecomposition, and it holds the following equation:

Where $\vec{v} \neq 0$.

The equation $A\vec{v} = \lambda \vec{v}$ can be written as the following:

Note: Identity matrix $I$ with $n \times n$ dimensions are multiplied with the constant $\lambda$ to perform matrix operations.

A trivial solution can be obtained by simply putting $\vec{v} = 0$; however, we are interested in getting a nontrivial solution. Hence, the following characteristic equation would be used to calculate the nonzero solution of $\vec{v}$.

It is interesting to note that the eigenspace equation revises the concept of nullspaces. It can be claimed that the eigenspace $E_\lambda(A)$ equates to the nullspace $N(A - \lambda I)$.

Note: To read more about eigenvalues and eigenvectors, click here.

Example 1

Calculate the eigenspace of the following matrix in $R^2$:

Using the characteristic equation as follows:

The operation leaves us with the following equation:

The determinant will be calculated as follows:

Solving the characteristic polynomial gives $\lambda = 1,4$. An eigenvector corresponding to each eigenvalue will be calculated.

For $\lambda = 1$, use the equation $(\lambda I - A) \vec{x} = 0$:

The following solution is obtained:

The answer can be written as follows:

The same procedure will be used to calculate the eigenvector for the eigenvalue $\lambda = 4$:

This equates to the following solution:

Hence, the eigenspace is formed by the eigenvalues $\lambda = 1,4$ with the corresponding vectors calculated above.

Example 2

Calculate the eigenspace of the following matrix in $R^3$:

Using the characteristic equation:

Calculating the determinant of the matrix leaves us with the following characteristic polynomial:

Solving the characteristic polynomial gives $\lambda = 2, 2, 9$. An eigenvector corresponding to each eigenvalue will be calculated.

For $\lambda = 2$,

The following solution is obtained:

The answer can be written as follows:

The same procedure will be used to calculate the eigenvector for the eigenvalue $\lambda = 9$:

This equates to the following solution:

Hence, the eigenspace is formed by the eigenvalues $\lambda = 2, 9$ with the corresponding vectors calculated above.

Quiz

Try the following questions for a better grip on the concept of eigenspaces:

Applications

Eigenvalues and eigenvectors have numerous applications in the real world, some of which have been listed below:

• Image processing: Eigenvectors are used to express the brightness of pixels in images used for facial recognition.

• Geology: Summarizes the orientation of the clast of a glacial till.

• Vibration analysis: Vibration modes of an object are depicted through eigenvectors.