From what I know (correct me if I am wrong):
$0$ as an eigen value of a real symmetric matrix implies it is Singular (Non- invertible).
I am not aware of any such property with reference to real symmetric matrices.
Also, I wish to know if the following statements are correct or not.
a) If two matrices have the same eigenvalues, they have the same eigenvectors. (I think it's false)
b) If two matrices have the same eigen vectors, they have the same eigen values. (I think that's true)