info:eu-repo/semantics/article
The spectral spread of Hermitian matrices
Fecha
2021-05Registro en:
Massey, Pedro Gustavo; Stojanoff, Demetrio; Zarate, Sebastian Gonzalo; The spectral spread of Hermitian matrices; Elsevier Science Inc; Linear Algebra and its Applications; 616; 5-2021; 19-44
0024-3795
CONICET Digital
CONICET
Autor
Massey, Pedro Gustavo
Stojanoff, Demetrio
Zarate, Sebastian Gonzalo
Resumen
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.