10.3.1 Theorem:
Let A be a real symmetric matrix and be distinct eigenvalues of A .Let
be nonzero such that
.                             , 1≤i≤k.
Then forms an orthonormal set.

                                                                                                                                                 

Proof of Theorem 10.3.1
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For i ≠ j, 1≤i, j≤k, since  = A, we have 
                                               
Since i ≠ j, we have