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Concrete minimal 3 x 3 Hermitian matrices and some general cases

Klobouk A. H., Varela A.

Demonstratio Mathematica

2017

Vol. 50, nr 1

330--350

Given a Hermitian matrix M ∈ M₃ (ℂ) we describe explicitly the real diagonal matrices Dм such that ║M + Dм║ ≤ ║M + D║ for all real diagonal matrices D ∈ M3(ℂ), where ║ · ║ denotes the operator norm. Moreover, we generalize our techniques to some n × n cases.

Existence and uniqueness of the solution in the frequency domain for the reflection-transmission problem in a viscoelastic layer

Caviglia G., Morro A.

Archives of Mechanics

2004

Vol. 56, nr 1

59-82

Existence and uniqueness for the reflection-transmission process originated in a viscoelastic solid layer are investigated. Wave propagation is framed within the Fourier-transform domain and the oblique incidence is modelled by a factor involving a transverse wave vector. The backward-forward propagation in the axial direction is ascertained through the sign of an energy flux. Next, a connection is established between the energy flux and an Hermitian matrix whose eigenvalues are half positive and half negative. The proof is given that if the matrix has two diagonal blocks, one of which is positive definite and the other is negative definite, the solution to the reflection-transmission problem exists and is unique. The condition on the blocks is found to hold, e.g., for obliquely propagating homogeneous waves in anisotropic elasticity or normally propagating waves in isotropic viscoelasticity.