My apology if the question is not appropriate. For me Eigenvectors are quite a mystery. Does it have any property that we can relate to the matrix it came from? By property I mean something like the sum of its elements, relation between real and imaginary part of the element, etc.
My specific interest is: consider a sample covariance matrix $A$ which is positive definite Hermitian matrix. Further,the sample covariance matrix is of size $N$ by $N$ and derived from an $N$ sensor system receiving band limited-signal over time.
Are there any specific properties of the eigenvectors of $A$? If there are any, how we can even formulate the problem? That Eigenvalues are variances and Eigenvectors are corresponding directions is a known property, but is there anything else?