Constructive characterization of Lipschitzian Q0-matrices

A Note on P1 - and Lipschitzian Matrices

The linear complementarity problem (q,A) with data A ∈ Rn×n and q ∈ Rn involves finding a nonnegative z ∈ Rn such that Az+ q ≥ 0 and zt(Az+ q) = 0. Cottle and Stone introduced the class of P1-matrices and showed that if A is in P1\Q, then K(A) (the set of all q for which (q,A) has a solution) is a half-space and (q,A) has a unique solution for every q in the interior of K(A). Extending the resu...

A Characterization of Constructive Dimension

In the context of Kolmogorov’s algorithmic approach to the foundations of probability, Martin-Löf defined the concept of an individual random sequence using the concept of a constructive measure 1 set. Alternate characterizations use constructive martingales and measures of impossibility. We prove a direct conversion of a constructive martingale into a measure of impossibility and vice versa, s...

A constructive arbitrary-degree Kronecker product decomposition of matrices

We propose a constructive algorithm, called the tensor-based Kronecker product (KP) singular value decomposition (TKPSVD), that decomposes an arbitrary real matrix A into a finite sum of KP terms with an arbitrary number of d factors, namely A = ∑R j=1 σj A dj ⊗ · · · ⊗A1j . The algorithm relies on reshaping and permuting the original matrix into a d-way tensor, after which its tensor-train ran...

Constructive Perturbation Theory for Matrices with Degenerate Eigenvalues

Abstract. Let A (ε) be an analytic square matrix and λ0 an eigenvalue of A (0) of multiplicity m ≥ 1. Then under the generic condition, ∂ ∂ε det (λI −A (ε)) |(ε,λ)=(0,λ0) 6= 0, we prove that the Jordan normal form of A (0) corresponding to the eigenvalue λ0 consists of a single m × m Jordan block, the perturbed eigenvalues near λ0 and their eigenvectors can be represented by a single convergent...

Derivative of Upper Lipschitzian