Some angularity and inertia theorems related to normal matrices

Abstract

Motivated by the definition of the inertia, introduced by Ostrowski and Schneider, a notion of angularity of a matrix is defined. The angularity characterizes the distribution of arguments of eigenvalues of a matrix. It is proved that if B and C are nonsingular matrices, then B ∗ AB and C ∗ AC have the same angularity provided they are diagonal. Some well-known inertia theorems (e.g. Sylvester's law) have been deduced as corollaries of this result. The case when C is permitted to be singular is discussed next. Finally, we prove that (a) any linear transformation T, on the set of n by n complex matrices, mapping Hermitian matrices into themselves and preserving the inertia of each Hermitian matrix is of the form T( A)= C ∗ AC or T( A)= C ∗L A′C where C is some nonsingular matrix, and (b) any linear transformation T mapping normal matrices into normal matrices and preserving the angularity of each normal matrix is also of one of the above forms, but with C= kU where k≠0 and U is unitary.