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How do you 'verify' the orthogonality of the eigenvectors of a matrix, let's say ${\bf A}$ , for example? I came across the result that a matrix ${\bf A}$ has orthogonal eigenvectors if ${\bf A^TA=AA^T}$ but is this a definitive test? How does such an equality verify orthogonality?

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I am not familiar with this result, but what you're saying is $A^tA = A A^t \Rightarrow A$ has orthogonal eigenvectors, but not necessarily vice versa. To see justification, it would be helpful to look at the proof. – andybenji Jan 3 '13 at 22:49

I suggest you read about Normal matrix

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