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′, instead of considering only one eigenvalue as traditional spectral theory does. Setting a=λ+λ′2≠0 and e=λ′−λ2>0, the information is delivered in geometric form, both metric and trigonometric, associated with various right-angled triangles exhibiting optimality properties quantified as ratios or product of |a| and e. The potential optimisation has a triple nature which offers two possibilities: in the case λλ′>0 they are characterised by e|a| and |a|e and in the case λλ′<0 by |a|e and |a|e. This nature is revealed by a key generalisation to indefinite matrices over R or C of Gustafson's operator trigonometry.