Hermitian matrices : Spectral coupling, plane geometry/trigonometry and optimisation

The paper presents the information processing that can be performed by a general hermitian matrix when two of its distinct eigenvalues are coupled, such as λ < λ′. Setting a = λ+λ ′ 2 and e = λ′−λ 2 > 0, the new spectral information that is provided by coupling is expressed in terms of the ratios e |a| (if λλ ′ > 0) or |a| e (if λλ ′ < 0) and of the product |a|e. The information is delivered in geometric form, both metric and trigonometric, associated with various right-angled triangles deriving from optimality conditions. The paper contains a generalisation to indefinite matrices over R or C of Gustafson’s operator trigonometry which in the matrix case assumes definiteness (mostly over R).