When the Eigenvectors of Matrix Always Orthogonal or Not (With Octave/Matlab Prove)

In Linear Algebra, an orthogonal matrix is defined as matrix with its transpose equal to its inverse,
$$ Q^T = Q^{-1}$$ or
$$ Q.Q^T = Q^T.Q = I $$ How about eigenvectors of Matrix? Are eigenvectors always orthogonal? The short answer is no. Eigenvectors of an arbitrary (but not degenerate) square real matrix A 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 $A^{T}A=AA^{T}$.

It follows that eigenvectors of a symmetric real matrix A (i.e $A =A^T$) are perpendicular. Another nice property of symmetric matrices is that their eigenvalues are real.

Let`s check using Octave/Matlab,