Mathematica Eterna
Open Access

Abstract

Characterizations and bounds for weighted sums of eigenvalues of normal and Hermitian matrices

Jorma K. Merikoski,Ravinder Kumar,

Let A ∈ C n×n be normal with eigenvalues λ1, . . . , λn, and let t1, . . . , tn ∈ C. It is well-known that max π∈Sn |t1λπ(1) + · · · + tnλπ(n) | = max n |t1u ∗ 1Au1 + · · · + tnu ∗ nAun| {u1, . . . , un} ⊂o C n o . Here Sn denotes the symmetric group of order n, and ⊂o means “is an orthonormal subset of . . . ”. If A is Hermitian and λ1 ≥ · · · ≥ λn, and if t1, . . . , tn ∈ R satisfy t1 ≥ · · · ≥ tn, then t1λ1 + · · · + tnλn = maxn t1u ∗ 1Au1 + · · · + tnu ∗ nAun | {u1, . . . , un} ⊂o C n o and tnλ1 + · · · + t1λn = min n t1u ∗ 1Au1 + · · · + tnu ∗ nAun | {u1, . . . , un} ⊂o C n o . We present bounds for the left-hand sides of all these equations by suitable choices of u1, . . . , un.

Published Date: 2018-09-05;