linear algebra — are eigenvectors orthogonal? 
Are eigenvectors always orthogonal? The short answer is no. Eigenvectors of an arbitrary (but not degenerate) square real matrix 
are sure to be independent (if there are no repeated eigenvalues), however they are not necessarily orthogonal.
Are eigenvectors of a symmetric matrix always orthogonal? The short answer is yes. Gilbert Strang gives the following definition: A real matrix has perpendicular eigenvectors if and only if 
. It follows that eigenvectors of a symmetric real matrix (i.e., 
) are perpendicular. Another nice property of symmetric matrices is that their eigenvalues are real.
Reference
Gilbert Strang, "Introduction to Linear Algebra, 4th Edition", Wellesley-Cambridge Press, 2009.
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