I have just learnt how to reduce a symmetric matrix to a diagonal one, using the equation below:
Where A is the original symmetric matrix and D is the diagonal matrix.
P is an orthogonal matrix whose columns consist of the normalised eigenvectors of A.
My problem is how to write the above equation in terms of A.
For example in the real numbers is we have
if we multiply on the left or the right by 2 we can still solve for x by the communicative property, but with matrices it is not so nice.
If we multiply on the left by the inverse of A we get
but if you multiply on the right you get
and since multiplication does not commute we cannot simplify.