Two matrix eigenvalue inequalities

Eigenvalue Inequalities and Schubert Calculus

Using techniques from algebraic topology we derive linear inequalities which relate the spectrum of a set of Hermitian matrices A1, . . . , Ar ∈ Cn×n with the spectrum of the sum A1 + · · ·+Ar. These extend eigenvalue inequalities due to Freede-Thompson and Horn for sums of eigenvalues of two Hermitian matrices.

Eigenvalue Problems Associated with Korn's Inequalities

Eigenvalue inequalities for graphs and convex subgraphs

For an induced subgraph S of a graph, we show that its Neumann eigenvalue λS can be lower-bounded by using the heat kernel Ht(x, y) of the subgraph. Namely, λS ≥ 1 2t ∑ x∈S inf y∈S Ht(x, y) √ dx √ dy where dx denotes the degree of the vertex x. In particular, we derive lower bounds of eigenvalues for convex subgraphs which consist of lattice points in an d-dimensional Riemannian manifolds M wit...

Approximate Antilinear Eigenvalue Problems and Related Inequalities

If T is a complex symmetric operator on a separable complex Hilbert space H, then the spectrum σ(|T |) of √ T ∗T can be characterized in terms of a certain approximate antilinear eigenvalue problem. This approach leads to a general inequality (applicable to any bounded operator T : H → H), in terms of the spectra of the selfadjoint operators ReT and ImT , restricting the possible location of el...

Isoperimetric Inequalities for a Nonlinear Eigenvalue Problem

An estimate for the spectrum of the two-dimensional eigenvalue problem Am + Xe" = 0 in D (X > 0), u = 0 on 3 D is derived, and upper and lower pointwise bounds for the solutions are constructed. 1. Let D be a simply connected bounded domain in the plane with a piecewise analytic boundary 3D. Consider the nonlinear Dirichlet problem Am(x) + Xe"w = 0 in D, (1) u(x) = 0 on dD, where X is a positiv...