Spectral refinement for clustered eigenvalues of quasi-diagonal matrices

On uniform connectivity of algebraic matrix sets

“…We will have that there is δ > 0 that can be chosen so that. 1] {|x − y| | x = y}} and p j (X j ) = p j (Ψ(X j )) = p j (Y j ) = 0 n , we will have that Y j =Ŵ X jŴ * = Ψ(X j ), otherwise we get a contradiction.…”

“…for each 1 ≤ j ≤ m. Furthermore, as a consequence of (3.37) and (3.39) we will have that, 1] is differentiable with respect to t, and satisfies the relation…”

“…It is enough to consider the case ε < 4 sin(1/8) < 1/2. Since max{ X j , Y j } ≤ 1 for each 1 ≤ j ≤ m, for the rest of the proof we will only consider the sets Z(p j ) ∩ [−1, 1], 1 ≤ j ≤ r. Let h p > 0 be a number chosen so that, 1] for each 1 ≤ j ≤ r, we have that h p ≤ 2. We will have that there is δ > 0 that can be chosen so that.…”

“…A common technique, implemented in order to achive uniformity in the approximation of the sequences mentioned before, consists in preconditioning each m-tuple X k ∈ M n k (C) m of the original sequence {X k } k≥1 to obtain a sequence {X k } k≥1 , with some desirable artificial properties. In these document we focus on spectral artificial properties, more specifically on eigenvalue clustering in the sense of [1,9,18].…”

Connecting Commuting Normal Matrices

“…By considering matrix paths as continuous/differentiable analogies of numerical linear algebra algorithms in the sense of [7], we will work on the adaptation and application of the results in sections §3 and §4 to the structure-preserving approximation/perturbation of families of structured matrices with some particular spectral behavior in the sense of [1,2,18,17,21].…”