The $D$-decomposition technique for linear matrix inequalities

In the framework of the theory of linear matrix inequalities, a method is proposed for determining all the domains in the parameter space having the property that an affine family of symmetric matrices has the same fixed number of like-sign eigenvalues inside each of the domains. The approach leans on the ideas of $D$-decomposition; it is particularly efficient in the problems involving few parameters. Generalizations of the method are considered along with its modifications to the presence of uncertainty.