Find an orthogonal matrix P and a diagonal matrix D such that
A = PDP^(-1).
I found P but how do I find D? Is it correct to do D = PAP^(-1)? Thanks in advance.
Although you could do , there is a much easier way:
In the process of finding , you probably found the eigenvalues and eigenvectors of . We know that if the eigenvalues of are , and each eigenvalue corresponds to an eigenvector such that are linearly independent, then the matrix will diagonalize , and . Now, it turns out that
Therefore, if you diagonalized with to get , then will be a diagonal matrix whose diagonal entries are the eigenvalues of , listed in the same order as you placed the eigenvectors of into .
Let me give a brief example:
Let
We may easily determine the eigenvalues of to be , and their corresponding eigenvectors:
Putting these eigenvectors into the columns of , we get
so for some diagonal . And if you find , you will find that it is
(try verifying this) and the entries along the main diagonal correspond exactly to the eigenvalues we found for .
But, if you were to change the order of the eigenvectors in , the entries in will also be in different order.
For example, if we had set
,
then we would get