A T is positive definite . D is a diagonal matrix. Thus Q T AQ is Lyapunov diagonally stable and by Lemma 1 preserves the inertia of diagonal matrices. G have the same sign. D is a rank one diagonal matrix.
T is also positive definite.
Consequently, the sign of trace Q T BQD is equal to the sign of the trace of the rank one diagonal matrix D . This condition is not sufficient as we can see in the following example. In general, the sum of matrices having the same inertia may have a different inertia. Inertia preserving matrices, SIAM J. Matrix Anal. Strongly inertia preserving matrices, SIAM J. Matrix Anal.