The spectral spread of Hermitian matrices

MASSEY, PEDRO; STOJANOFF, DEMETRIO; ZÁRATE, SEBASTIÁN

LINEAR ALGEBRA AND ITS APPLICATIONS

ELSEVIER SCIENCE INC

Año: 2021 vol. 616 p. 19 - 44

0024-3795

Let A be an n×n complex Hermitian matrix and let λ(A)=(λ1,?,λn)∈Rn denote the eigenvalues of A, counting multiplicities and arranged in non-increasing order. Motivated by problems arising in the theory of low rank matrix approximation, we study the spectral spread of A, denoted Spr+(A), given by Spr+(A)=(λ1−λn,λ2−λn−1,?,λk−λn−k+1)∈Rk, where k=[n/2] (integer part). The spectral spread is a vector-valued measure of dispersion of the spectrum of A, that allows one to obtain several submajorization inequalities. In the present work we obtain inequalities that are related to Tao´s inequality for anti-diagonal blocks of positive semidefinite matrices, Zhan´s inequalities for the singular values of differences of positive semidefinite matrices, extremal properties of direct rotations between subspaces, generalized commutators and distances between matrices in the unitary orbit of a Hermitian matrix.