Assume that we have two real symmetric matrices A and B, where A is a positive diagonal matrix, and B is a symmetric matrix with one eigenvalue λ = 0. Assume that H= AB; is it possible to proof that the matrix H has the same properties than B (semi-positive, negative or indefinite matrix)? That is means the eigenvalues of H have the same signe than the eigenvalues of B.

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The eigenvalues of $H$ are the same as the eigenvalues of $A^{1/2} B A^{1/2}$. See Sylvester's Law of Inertia.