I have been looking at the solution to a question and I don't understand how the eigenstates are calculated. The question concerns a 3-state spin-1-system with angular momentum l=1. The 3 eigenstates of L(adsbygoogle = window.adsbygoogle || []).push({}); _{3}are given as ## \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} ## , ## \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} ## , ## \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} ## which leads to the z-component of angular momentum as L_{3}= ## \hbar ## ## \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -1 \end{pmatrix} ##. When I try to calculate the eigenvalues and eigenvectors of the L_{3}matrix using determinants I get no answer. Can anybody tell me how to get the eigenvalues and eigenvectors ? Thanks

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# Eigenstates in a 3-state spin 1 system

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