This one is a little confusing to me, but I'll show what I've done so far.
I know that a symmetric matrix is a matrix that is equal to its transpose, like the identity matrix.
I also know that for a vector to be an eigenvector of some matrix $A$, the following must be true
$Av = \lambda v$,
and for the eigenvectors to be orthogonal, their dot product must be $0$.
Does this mean that the only eigenvectors for a symmetric $n\times n$ matrix are the zero vectors?
Any help will be appreciated.