how to find the distinct eigenvectors from a repeated eigenvalue

Question

As the title, how to find the distinct eigenvectors of an certain eigenvalue if the algebraic multiplicity of that eigenvalue is not 1? The target matrix is real and symmetric. I know these eigenvectors spans a subspace, but I have no idea how to find each of them. I have tried Gram–Schmidt method by first guessing an eigenvector and then finding rest of them, but it does not work. Can someone help me?

Answer 1

In the case of real symmetric matrices, the algebraic multiplicity of the eigenvalue $\lambda$ is equal to the dimension of the kernel $\ker (A-\lambda I)$. Therefore, once you get the eigenvalue, just take any basis for $\ker(A-\lambda I)$ and you will get "enough" vectors.