A real matrix A is inertia preserving if in $AD = \operatorname{in} D$, for every invertible diagonal matrix D. This class of matrices is a subset of the D-stable matrices and contains the diagonally stable matrices.In order to study inertia-preserving matrices, matrices that have no imaginary eigenvalues are characterized. This is used to characterize D-stability of stable matrices. It is also shown that irreducible, acyclic D-stable matrices are inertia preserving.