Given a matrix we can find it eigen value on a given basis set . Suppose the eigen vector is

Put in the eigen equation

We can act on the left, but in general, the basis set are not orthogonal.

This is the General Eigen Value Problem.

One way to solve the problem, is the “reconfigure” the basis so that it is orthogonal. However, in computation, non-orthogonal basis could give supreme advantage. So, the other way is split the problem. First solve the ,

The eigen system of is

Here, is a diagonal matrix of eigen value. Now, we define a new non-unitary matrix

Notices that

Thus,

We know that, the form is a transform that from one basis to another basis, i.e.

and for any operator,

We put this back to the general problem

Thus, we can solve the , get the eigen system, then use

For example,

The eigen system is

The matrix is

We can verify that

Let a Hamilton is

The Hamiltonian in the basis is

The eigen values in the S frame are

which is not the correct eigen values.

Now, we transform the Hamiltonian into orthogonal basis

The eigen values are

where are the correct pair. The un-normalized eigen vector are

We can verify that

### Like this:

Like Loading...

## Leave a Reply