LAUSR

Abstract 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 ′ − λ 2 > 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 λ λ ′ > 0 they are characterised by e | a | and | a | e and in the case λ λ ′ 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.