Approximating Coalescing Points for Eigenvalues of Hermitian Matrices of Three Parameters

We consider a Hermitian matrix valued function A(x) ∈ C, smoothly depending on parameters x ∈ Ω ⊂ R, where Ω is an open bounded region of R. We develop an algorithm to locate parameter values where the eigenvalues of A coalesce. We give theoretical justification and implementation details. Finally, we illustrate the technique on several problems. Notation. With A ∈ Cn×n we indicate an n×n complex matrix, and A∗ is the Hermitian of A: A∗ = ĀT . By deafult, we consider the 2-norm for vectors and the induced norm on matrices, though at times we will use also the Frobenius norm ‖ · ‖F for matrices, ‖A‖F = trace(A∗A). A matrix valued function A : R → Cn×n, continuous with its first k derivatives (k ≥ 0), is indicated as A ∈ Ck(R,Cn×n); if k = 0, we also simply write A ∈ C, and in the analytic case we will write A ∈ Cω. If A ∈ Ck(R,Cn×n) is periodic of (minimal) period τ > 0, we write it as A ∈ Ck τ (R,Cn×n). With Ω ⊂ R3 we indicate an open bounded region of R3 diffeomorphic to the open unit ball, and with x = (x1, x2, x3) we indicate coordinates in Ω. The surface bounding Ω will be denoted with S, and it will therefore be diffeomorphic to the standard 2-sphere. The standard unit vectors are denoted e1, e2, e3. For a function A(x), x ∈ Ω, we will write A ∈ Ck(Ω,Cn×n) as appropriate.