Prerequisites:
For a given matrix
A the set of all eigenvectors of
A associated with an eigenvalue
spans a subspace, which is called the
Eigenspace of
A with respect to
and is denoted by
. The set of all eigenvalues of
A is called
Eigenspectrum, or just spectrum, of
A.
If
is an eigenvalue of A, then the corresponding eigenspace
is the solution space of the homogeneous system of linear equations
. 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 possess the same eigenvalues but not necessarily the same eigenvectors.The eigenspace is the null space of since
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
and
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.