Related papers: Structured matrices in the application of bivariat…

In this paper, we introduce two new forms of the dual Hartwig-Spindelb{\"o}ck decomposition and employ them to derive explicit representations for several classes of dual generalized inverses. Building on these representations, we further…

A matrix-compression algorithm is derived from a novel isogenic block decomposition for square matrices. The resulting compression and inflation operations possess strong functorial and spectral-permanence properties. The basic observation…

Theory of matrix factorizations is useful to study hypersurfaces in commutative algebra. To study noncommutative hypersurfaces, which are important objects of study in noncommutative algebraic geometry, we introduce a notion of…

We consider the problem of recovering a unitary eigendecomposition of a complex unitary matrix from that of its embedded real-valued formulation. Such formulations arise naturally in scientific computing workflows that employ…

Using a lemma of Davis on Gram matrices applied to the classical Orthogonal Polynomials to generate reproducing kernel interpolation over the classical domains for polynomials. These kernels have terms which are exact over the rational…

In this paper we outline a Matrix Ansatz approach to some problems of combinatorial enumeration. The idea is that many interesting quantities can be expressed in terms of products of matrices, where the matrices obey certain relations. We…

We study implicit regularization when optimizing an underdetermined quadratic objective over a matrix $X$ with gradient descent on a factorization of $X$. We conjecture and provide empirical and theoretical evidence that with small enough…

We consider rank-one non-symmetric tensor estimation and derive simple formulas for the mutual information. We start by the order 2 problem, namely matrix factorization. We treat it completely in a simpler fashion than previous proofs using…

Using three different approaches, we analyze the complexity of various birational maps constructed from simple operations (inversions) on square matrices of arbitrary size. The first approach consists in the study of the images of lines,…

We present a method for randomizing formulas for bilinear computation of matrix products. We consider the implications of such randomization when there are two sources of error: One due to the formula itself only being approximately…

We study computable embeddings for pairs of structures, i.e. for classes containing precisely two non-isomorphic structures. Surprisingly, even for some pairs of simple linear orders, computable embeddings induce a non-trivial degree…

We discuss some applications of an intrinsic multipication in the space of simple loops in a surface.

In this note we show that the degree of the interpolation polynomial for equidistant base points is characterized by the regularity of matrices of combinatorical type.

Vlastimil Pt\'ak's method of nondiscrete induction is based on the idea that in the analysis of iterative processes one should aim at rates of convergence as functions rather than just numbers, because functions may give convergence…

We give new improvements to the Chudnovsky-Chudnovsky method that provides upper bounds on the bilinear complexity of multiplication in extensions of finite fields through interpolation on algebraic curves. Our approach features three…

We introduce an arithmetic version of the multivariate Tutte polynomial, and (for representable arithmetic matroids) a quasi-polynomial that interpolates between the two. A generalized Fortuin-Kasteleyn representation with applications to…

In this paper the main results in arXiv:0901.3179v3, related to the matrix representation of polynomial maps, are restated in traditional way of linear algebra assuming that variable vectors are presented as column vectors. Some new results…

Convenient parameterizations of matrices in terms of vectors transform (certain classes of) matrix equations into covariant (hence rotation-invariant) vector equations. Certain recently introduced such parameterizations are tersely…

We study and derive algorithms for nonlinear eigenvalue problems, where the system matrix depends on the eigenvector, or several eigenvectors (or their corresponding invariant subspace). The algorithms are derived from an implicit…

We present a detailed study of the combinatorial interpretation of matrix integrals, including the examples of tessellations of arbitrary genera, and loop models on random surfaces. After reviewing their methods of solution, we apply these…