Characteristic Equation
Computing Eigenvalues
In the previous section, we use the applet to find the eigenvectors and eigenvalues of any 2 x 2 matrix. In fact, we can compute them in a systematic way, as illustrated in the following example:
Example: Let . Find all the eigenvalues and their eigenvectors.
By definition, if is an eigenvector corresponding to the eigenvalue (yet to be determined), we have . We can rewrite this equation as follows:
It becomes a homogeneous equation and an eigenvector is exactly its non-trivial solution. Recall that such a homogeneous equation has non-trivial solutions if and only if . It turns out that the determinant can be expressed as a polynomial in . It is called the characteristic polynomial of and the polynomial equation is called the characteristic equation of .
In our example, we write down the characteristic equation explicitly as follows:
Computing the 2 x 2 determinant, we get
Hence, the eigenvalues of are and .
Computing Eigenvectors
To find the eigenvectors corresponding to each eigenvalue, we just need to put the eigenvalue into the homogeneous equation and solve for the solutions:
When ,
It can easily be row reduced into the following: . Hence, we have
for any real number . The set of all these eigenvectors and the zero vector form the subspace , which is called the eigenspace of corresponding to .
When ,
It can easily be row reduced into the following: . Hence, we have
for any real number . The set of all these eigenvectors and the zero vector form the subspace , which is the eigenspace of corresponding to .
Characteristic Equation of an n x n Matrix
Given an n x n matrix , we can find its eigenvalues by the same method: Solve the characteristic equation . It is a polynomial equation in of degree . Therefore, there are at most distinct eigenvalues of .
Example: Find the characteristic equation of .
Solution:
Since it is an upper triangular matrix, the determinant is the product of all diagonal entries i.e.
Therefore, the eigenvalues of are and . The eigenvalue is said to have multiplicity 2 because the factor appears twice in the characteristic equation. In general the multiplicity of an eigenvalue is its multiplicity as a root of the characteristic equation.
Let . Find all the eigenvalue(s) and their eigenvectors if exist.
Let . Find all the eigenvalues and their eigenvectors if exist.