The statement is true, however I would like to know what does it mean geometrically. How do I check that the eigen vectors are orthogonal eigen vectors?
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1Does this answer your question? [Eigenvectors of real symmetric matrices are orthogonal](https://math.stackexchange.com/questions/82467/eigenvectorsofrealsymmetricmatricesareorthogonal) – Rick Jun 13 '20 at 11:03

I am looking for a geometric reason @Rick – Antimony Jun 13 '20 at 11:05

1Eigenvectors beonging to *different* eigenvalues are orthogonal to each other (if $A^t=A$). Eigenvectors belonging to the same eigenvalue, even if linearly independent of each other, need not be orthogonal to each other. – Gerry Myerson Jun 13 '20 at 11:18

1The trouble with your question, smita, is that *orthogonality* is not a geometric concept. Sure, it's based on the geometric concept of a right angle in two and threedimensional Euclidean space, but outside of that context, in higher dimensions and in abstract vector spaces, there is no concept of right angles or of orthogonality beyond what we get from algebra, from defining an inner product and calling two vectors orthogonal if their inner product is zero. The eigenvector statement is purely algebraic; it has geometric content only by virtue of interpreting the algebra. – Gerry Myerson Jun 14 '20 at 00:10

You may find some of the answers at https://math.stackexchange.com/questions/37398/whatisthegeometricinterpretationofthetranspose helpful. – Gerry Myerson Jun 14 '20 at 00:16