(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

The matrix representation of an operator is

[tex]

\left(

\begin{array}{cccc}

1 & 0 & a & 0\\

0 & 1 & 0 & b\\

a & 0 & 1 & 0\\

0 & b & 0 & 1\\

\end{array}

\right)

[/tex]

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

3. The attempt at a solution

I can easily show that [tex] \frac{1}{\sqrt{2}} \left( \begin{array}{cccc} 1 & 0 & 1 & 0 \end{array} \right) [/tex] is the eigenstate of the operator and found that it's eigenvalue is [tex] 1+a [/tex]. 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

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# Matrix Operator Question

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