How Do You Determine Eigenstates and Eigenvalues for a Given Matrix Operator?

[tarq]

Homework Statement

The matrix representation of an operator is

$$\left( \begin{array}{cccc} 1 & 0 & a & 0\\ 0 & 1 & 0 & b\\ a & 0 & 1 & 0\\ 0 & b & 0 & 1\\ \end{array} \right)$$

Show that $$\frac{1}{\sqrt{2}} \left( \begin{array}{cccc}1 & 0 & 1 & 0\end{array} \right)$$ is an eigenstate of the operator and derive its eigenvalue. Give one other eigenstate of the operator with its eigenvalue.

The Attempt at a Solution

I can easily show that $$\frac{1}{\sqrt{2}} \left( \begin{array}{cccc} 1 & 0 & 1 & 0 \end{array} \right)$$ is the eigenstate of the operator and found that it's eigenvalue is $$1+a$$. I however don't understand how to work out the other eigenstates of the operator. Do you have to simply guess the other eigenstates and then check to see if they are correct by checking if it fulfills the condition [Operator][eigenstate]=[eigenvalue][eigenstate] ?, thanks for the help

 

[phsopher]

Well there is a procedure for finding eigenvalues and eigenstates. The eigenvalue equation is Ax=ax <=> (A-aI)x=0 where I is the identity matrix. This has non-trivial solutions if the matrix (A-aI) is not invertible. So the condition for the existence of eigenvalues is det(A-aI)=0 and they are found as the roots of this equation. Once you know the eigenvalues you can find the corresponding eigenvectors by plugging a general vector into the eigenstate equation and solving for the components of the vector.

However, in your case it easy to guess another eigenstate just by looking at the matrix and the given eigenstate. That is obviously a lot easier and sufficient since you aren't asked to derive the vectors.