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The eigenvectors ($\psi _i$) of the following matrix are rather complicated $$ \rho =\left( \begin{array}{cccc} G_+ & k & k & G_- \\ k & L_+ & L_- & k \\ k & L_- & L_+ & k \\ G_- & k & k & G_+ \\ \end{array} \right) $$ So, I want to suppose that $ E_i $ is the eigenvalue of $\rho$ that corresponding to the eigencfunction $\psi _i$, then to evaluate ($\psi _i$) as a function of $E_i $.

Is there a way I can do this with Mathematica?

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  • $\begingroup$ Is your matrix purely symbolic? Or do the variables have numeric values? $\endgroup$
    – MikeY
    Commented Jun 10, 2020 at 19:01

1 Answer 1

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Assume your matrix is

ρ ={{p, k, k, m},
    {k, a, b, k},
    {k, b, a, k},
    {m, k, k, p}};

and you have found eigenvalues using

Eigenvalues[ρ]

And for some reason, you do not like Eigensystem. Then you can use the NullSpace command

NullSpace[ρ - (a - b) IdentityMatrix[4]]
NullSpace[ρ - (p - m) IdentityMatrix[4]]

to recover the eigenvectors corresponding to eigenvalues $E_1=a-b$, and $E_2=p-m$, etc.

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