Calculating Eigenkets from Matrix w/ Orthonormal Basis

mkbh_10
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How am i supposed to write eigenkets of an operator whose matrix is given to me given that the two ket vectors form an orthonormal basis .
 
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Eigenkets is just the "quantum mechanical" name for eigenvectors. So you find the eigenvectors, and then make them orthonormal (e.g. apply the Gram-Schmidt orthonormalisation process)
 

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