finding eigenvalues
This example investigates eigenvalues and the similarity transformation used
to diagonalize matrices. We seek the eigenvalues of the matrix below. Afterward,
we can transform this matrix into a diagonal matrix which has many useful applications.
Here, we need to solve the corresponding matrix equation;
or
rearranging gives
or
We seek the values for and .
First, we need to solve the characteristic equation of . We do this by finding
. First, calculating gives;
Next, calculating yields
Substituting into gives…
so that and the corresponding eigenvector is
where
Substituting gives…
so that and the corresponding eigenvector is
where
Finally, to diagonalize we let the eigenvectors be the columns of a new matrix
and then since our eigenvectors are linearly independent we can also find;
then we create a diagonal matrix as follows…
Computing powers of is a very useful application of . Solving for lets us compute powers of
so that
or
Title | finding eigenvalues |
---|---|
Canonical name | FindingEigenvalues |
Date of creation | 2013-03-22 15:52:35 |
Last modified on | 2013-03-22 15:52:35 |
Owner | PrimeFan (13766) |
Last modified by | PrimeFan (13766) |
Numerical id | 6 |
Author | PrimeFan (13766) |
Entry type | Example |
Classification | msc 15A18 |