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I have the following matrix \begin{equation} A= \begin{pmatrix} 0 &1  &0  &0  \\ 1 & 0 &0 &0\\ 0 &0&0&1\\ 0&0&1&0 \end{pmatrix} \end{equation} The eigenvalues of this matrix is $\left\{1,1,-1,-1\right\}$. We set \begin{equation} |\psi\rangle=\alpha|++\rangle + \beta |+-\rangle +\gamma |-+\rangle +\delta|--\rangle \end{equation} To determine the constants $\left\{\alpha,\beta,\gamma,\delta\right\}$, we apply the equation to the eigenvalues for the  positive energy firstly as  \begin{equation} \begin{pmatrix} 0 &1  &0  &0  \\ 1 & 0 &0 &0\\ 0 &0&0&1\\ 0&0&1&0 \end{pmatrix}\begin{pmatrix} \alpha \\ \beta\\ \gamma\\ \delta \end{pmatrix}=\begin{pmatrix} \alpha \\ \beta\\ \gamma\\ \delta \end{pmatrix} \end{equation} which implies that $\beta=\alpha$, $\delta=\gamma$. The normalization condition gives \begin{equation} \langle \psi \lvert\psi \rangle=1 \longrightarrow 2\rvert\alpha\rvert^{2} + 2\rvert\gamma\rvert^{2}=1 \end{equation} My problem is that: I did note know the method used to find the relation between $\alpha$ et $\gamma$ and to do the rest of the calculation.

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You have found the eigenvectors in the 1-eigenspace to be $$ |\psi \rangle = \alpha \begin{pmatrix} 1\\ 1\\ 0\\ 0 \end{pmatrix} + \gamma \begin{pmatrix} 0\\ 0\\ 1\\ 1 \end{pmatrix} = \alpha |v_1\rangle + \gamma|v_2\rangle, $$ where $\alpha$ and $\gamma$ are any complex numbers constrained by $|\alpha|^2 + |\gamma|^2 = \frac{1}{2}$. This means that this eigenspace is two-dimensional, and any linear combination of the basis vectors $|v_1\rangle$ and $|v_2\rangle$ is an eigenstate of your matrix $A$ with eigenvalue $1$. In other words, you are completely free to choose whichever linear combination in the subspace you want, and you have to provide an extra constraint to fix the answer to whatever question you have.

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  • $\begingroup$ I'm not sure if I understand your question. Do you mean 'how will we know the states in which we express the operator $\hat{A}$, that is, $|++\rangle, |+-\rangle,|-+\rangle,|--\rangle$ are normalized or not'? That is up to your definition, you can define them as normalized states at the beginning. $\endgroup$
    – lazerbeam
    Apr 15, 2021 at 19:41

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