This example investigates eigenvalues



and the similarity transformation
used
to diagonalize matrices. We seek the eigenvalues of the matrix A 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 λ and X.
First, we need to solve the characteristic equation

of A. We do this by finding
det(A-λI). First, calculating A-λI gives;
Next, calculating det(A-λI) yields
|
det(A-λI)=(2-λ)2-1=λ2-4λ+3=(λ-1)(λ-3)=0 |
|
Substituting λ=1 into (A-λI)X gives…
so that x2=-x1 and the corresponding eigenvector


is
where t≠0.
Substituting λ=3 gives…
so that x2=x1 and the corresponding eigenvector is
where t≠0.
Finally, to diagonalize A 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 A is a very useful application of D. Solving for A lets us compute powers of A
so that
or