Eigenspace and Eigenspectrum Values in a Matrix

Prerequisites:

For a given matrix A the set of all eigenvectors of A associated with an eigenvalue $\lambda$ spans a subspace, which is called the Eigenspace of A with respect to $\lambda$ and is denoted by $E_\lambda$ . The set of all eigenvalues of A is called Eigenspectrum, or just spectrum, of A.
If $\lambda$ is an eigenvalue of A, then the corresponding eigenspace $E_\lambda$ is the solution space of the homogeneous system of linear equations $(A-\lambda I)x = 0$ . Geometrically, the eigenvector corresponding to a non – zero eigenvalue points in a direction that is stretched by the linear mapping. The eigenvalue is the factor by which it is stretched. If the eigenvalue is negative, then the direction of the stretching is flipped.

Below are some useful properties of eigenvalues and eigenvectors in addition to the properties which are already listed in the article Mathematics | Eigen Values and Eigen Vectors.

A matrix A and its transpose $A^{T}$ possess the same eigenvalues but not necessarily the same eigenvectors.

The eigenspace $E_\lambda$ is the null space of $A - \lambda I$ since
$Ax = \lambda x$ $\Longleftrightarrow$ $Ax - \lambda x = 0$ $\Longleftrightarrow$ $(A - \lambda I)x = 0$ $\Longleftrightarrow$ $x\in ker(A - \lambda I)$

Note: ker stands for Kernel which is another name for null space.

Computing Eigenvalues, Eigenvectors, and Eigenspaces:

Consider given 2 X 2 matrix:


Step 1: Characteristic polynomial and Eigenvalues.
The characteristic polynomial is given by 
det() 





After we factorize the characteristic polynomial, we will get



which gives eigenvalues as  and 

Step 2: Eigenvectors and Eigenspaces
We find the eigenvectors that correspond to these eigenvalues by looking 
at vectors x such that 
 


For  we obtain



After solving the above homogeneous system of equations,
we will obtain a solution space



This eigenspace is one dimensional as it possesses a single basis vector.
Similarly, we find eigenvector for  by solving
the homogeneous system of equations



This means any vector , where  
such as  is an eigenvector with 
eigenvalue 2. This means eigenspace is given as

The two eigenspaces $E_5$ and $E_2$ in the above example are one dimensional as they are each spanned by a single vector. However, in other cases, we may have multiple identical eigenvectors and the eigenspaces may have more than one dimension.

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Last Updated : 15 Jan, 2020

Eigenspace and Eigenspectrum Values in a Matrix

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