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From the property of determinant: det (AB) =det (A) â det (B), it is easy to see that det (A) =det (Gâ 1 âÎ âG) =det (Gâ 1) âdet (Î ) âdet (G) =det (Gâ 1 G) âdet (Î ) =det (Î ) = â i=1 N λi, where G is the square NöN matrix whose ith column is the basis eigenvector gi of A, and Î is the diagonal matrix whose diagonal elements are the corresponding eigenvalues, i.e., λi = Î ii.
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From the property of determinant: det (AB) =det (A) âdet (B), it is easy to see that det (A) =det (Gâ 1 âÎ âG) =det (Gâ 1) âdet (Î ) âdet (G) =det (Gâ 1 G) âdet (Î ) =det (Î ) = â i=1 N λi, where G is the square NöN matrix whose ith column is the basis eigenvector gi of A, and Î is the diagonal matrix whose diagonal elements are the corresponding eigenvalues, i.e., λi = Î ii.
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