# If I have the eigenvalues, can I find the corresponding eigenvectors?

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?

• Is your matrix purely symbolic? Or do the variables have numeric values? Jun 10, 2020 at 19:01

## 1 Answer

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.