G is equal to trace TG . We conclude the paper by characterizing matrices that preserve the inertia of the real symmetric matrices and the matrices that preserve the inertia of the Hermitian matrices. 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.