Factorizations of totally positive matrices

Abstract

Different approaches to the decomposition of a nonsingular totally positive matrix as a product of bidiagonal matrices are studied. Special attention is paid to the interpretation of the factorization in terms of the Neville elimination process of the matrix and in terms of corner cutting algorithms of Computer Aided Geometric Design. Conditions of uniqueness for the decomposition are also given.

On best rank n matrix approximations

“…For an explanation and history of this and similar results, see Pinkus [10,Chapter 3]. In addition, we have this next result which follows from Theorem 3.3 (a).…”

Total Positivity in Stable Semigroups

“…This can be viewed as a generalization of the classical Cauchy kernel K 1/2 , whose total positivity of infinite order serves as a basic example in [8] p. 149 and is a consequence of Cauchy's double alternant formula-see also Example 4.3 in [11]. As a consequence of the above result, we prove the following nontrivial characterization.…”

“…One says that K is TP ∞ if these inequalities hold for all n. We refer to [8] for the classic account on this field and its numerous connections with analysis. Let us also mention the recent monograph [11] for a more linear algebraic point of view and updated references, and the survey paper [5] for other algebraic applications of total positivity. General results on the total positivity in space-time of Markovian semigroups are given in [8] pp.…”

“…115-117. Observe also the connection with KatkovaVishnyakova's general criterion on 2 × 2 minors-see Theorem 2.16 in [11], which is however too stringent in our framework. (b) The main result of [18] entails the further equivalence…”

On discrete ℓ1-regularization

“…These general remarks are significantly refined when A is a strictly totally positive (STP) matrix. The importance of STP matrices is well-established [5,14]. Under this setting, the relationship between the number of nonzero coordinates of a distinguished solution of the DLR problem is precisely explained as a function of the regularization parameter for a certain class of vectors in R m .…”