Related papers: The Choi-Cholesky algorithm for completely positiv…
In this paper, we prove that if the matrix of the linear system is symetric, the Cholesky decomposition can be obtained from the Gauss elimination method without pivoting, without proving that the matrix of the system is positive definite.
This study investigates Hermitian linear maps, focusing on their decomposition into completely positive (CP) maps and their extensions to CP maps using auxiliary spaces. We derive a precise lower bound on the Hilbert-Schmidt norm of the…
We consider the problem of writing an arbitrary symmetric matrix as the difference of two positive semidefinite matrices. We start with simple ideas such as eigenvalue decomposition. Then, we develop a simple adaptation of the Cholesky that…
An alternative, geometrical proof of a known theorem concerning the decomposition of positive maps of the matrix algebra $M_{2}(\mathbb{C})$ has been presented. The premise of the proof is the identification of positive maps with operators…
Let $X^{(2)}$ denote the second symmetric product space of a partially ordered vector space $X$, endowed with the projective cone. A characterization of linear maps $T\colon X^{(2)}\to X^{(2)}$ which preserve the set of all positive…
We introduce a real-parameter refinement of the classical integer hierarchies underlying Schmidt number, block-positivity, and $k$-positivity for maps between matrix algebras. Starting from a compact family of $\alpha$-admissible unit…
The modified Cholesky decomposition is commonly used for precision matrix estimation given a specified order of random variables. However, the order of variables is often not available or cannot be pre-determined. In this work, we propose…
A new runtime environment for the execution of recursive matrix algorithms on a supercomputer with distributed memory is proposed. It is designed both for dense and sparse matrices. The environment ensures decentralized control of the…
The description of the dynamics of an open quantum system in the presence of initial correlations with the environment needs different mathematical tools than the standard approach to reduced dynamics, which is based on the use of a…
The phenomenological dissipation of the Bloch equations is reexamined in the context of completely positive maps. Such maps occur if the dissipation arises from a reduction of a unitary evolution of a system coupled to a reservoir. In such…
We introduce the geometric mean and the parallel sum of completely positive (CP) maps on von Neumann algebras, based on the Pusz--Woronowicz theory of positive sesquilinear forms. We provide a concrete characterization via a block matrix…
Many neural learning algorithms require to solve large least square systems in order to obtain synaptic weights. Moore-Penrose inverse matrices allow for solving such systems, even with rank deficiency, and they provide minimum-norm vectors…
Linear models have found widespread use in statistical investigations. For every linear model there exists a matrix representation for which the ReML (Restricted Maximum Likelihood) can be constructed from the elements of the corresponding…
Maps that are not completely positive (CP) are often useful to describe the dynamics of open systems. An apparent violation of complete positivity can occur because there are prior correlations of the principal system with the environment,…
We present a general characterization of k-positivity for a positive map in terms of the estimation of the Ky Fan norm of the matrix constructed from the Kraus operators of the associated completely positive map. Combining this with the…
Kraus maps (completely positive trace preserving maps) arise classically in quantum information, as they describe the evolution of noncommutative probability measures. We introduce tropical analogues of Kraus maps, obtained by replacing the…
In this paper we provide an algorithm, similar to the simplex algorithm, which determines a rational cp-factorization of a given matrix, whenever the matrix allows such a factorization. This algorithm can be used to show that every integral…
Inspired by the seminal work of Andr\'e-Louis Cholesky -- whose contributions remain crucial in broader sciences even after more than a century -- Cooper, Hanna and Whitlatch (2024) developed a theory of positive matrices over finite…
Working in any model theoretic structure, we single out a class of definable bipartite graphs that admit definable, close to perfect matchings. We use this result to prove a strengthening of Tarski's theorem for the definable setting.
For two positive maps $\phi_i:B(\mathcal{K}_i)\to B(\mathcal{H}_i)$, $i=1,2$, we construct a new linear map $\phi:B(\mathcal{H})\to B(\mathcal{K})$, where $\mathcal{K}=\mathcal{K}_1\oplus\mathcal{K}_2\oplus\mathbb{C}$,…