Global Properties of Eigenvalues of Parametric Rank One Perturbations for Unstructured and Structured Matrices

Abstract

General properties of eigenvalues of $$A+\tau uv^*$$ A + τ u v ∗ as functions of $$\tau \in {\mathbb {C} }$$ τ ∈ C or $$\tau \in {\mathbb {R} }$$ τ ∈ R or $$\tau ={{\,\mathrm{{e}}\,}}^{{{\,\mathrm{{i}}\,}}\theta }$$ τ = e i θ on the unit circle are considered. In particular, the problem of existence of global analytic formulas for eigenvalues is addressed. Furthermore, the limits of eigenvalues with $$\tau \rightarrow \infty $$ τ → ∞ are discussed in detail. The following classes of matrices are considered: complex (without additional structure), real (without additional structure), complex H-selfadjoint and real J-Hamiltonian.

A note on a conjecture concerning rank one perturbations of singular M-matrices

“…Consider A = ρ(H)I 4 − H, and B(t) = A + tvw . Following [5], see also [6], we have that for t → ∞ the eigenvalues of B(t) behave as follows: one is positive, and approximately equal to tw v + O(1), and the other three converge to the roots of the polynomial p vw (λ) = w m A (λ)(λI 4 − A) −1 v, where m A (λ) is the minimal polynomial of A. In this case, p vw (λ) is given by…”

“…Hence det A = 0 and trace A = µ. Then from [6] and [5] we have that the eigenvalues of B(t) which are not eigenvalues of A are the solutions of w (λI…”

A note on a conjecture concerning rank one perturbations of singular M-matrices

“…Hence det A = 0 and trace A = µ. Then from [13], Proposition 2 (see also [12], Proposition 2.2) we have that the eigenvalues of B(t) which are not eigenvalues of A are the solutions of w (λI 2 − A) −1 v = 1 t . Multiplying left and right with the characteristic polynomial p A (λ) of A, one sees that this is equivalent to λ being a solution of…”

“…Mostly, however, only the behaviour for small values of t has been studied ( [8,16], see also [5,10] for more detailed analysis). The problem of considering the behaviour of the eigenvalues for large values of t was considered in [12,13]. Restrictions on A, v and w, allowing only certain structured matrices, where considered in e.g.…”

“…Then H is nonnegative and irreducible, and the eigenvalues of H are 0, 0.1±0.1i, 0. It should be noted that the statements in [12], Theorem 4.1 are made for generic vectors v and w only. However, the fact that v and w have positive entries allows us to apply the results of that theorem in this particular case, which can be seen by a close inspection of the proofs in [12], Lemma 2.1 and Theorem 4.1.…”