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Sylvester's law of inertia is a theorem in matrix algebra about certain properties of the coefficient matrix of a real quadratic form that remain invariant under a change of basis. Namely, if is a symmetric matrix, then for any invertible matrix , the number of positive, negative and zero eigenvalues of is constant. This result is particularly useful when is diagonal, as the inertia of a diagonal matrix can easily be obtained by looking at the sign of its diagonal elements.
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