Related papers: Matrix decompositions over the double numbers
A general theorem on factorization of matrices with polynomial entries is proven and it is used to reduce polynomial Darboux matrices to linear ones. Some new examples of linear Darboux matrices are discussed.
Motivated by a certain molecular reconstruction methodology in cryo-electron microscopy, we consider the problem of solving a linear system with two unknown orthogonal matrices, which is a generalization of the well-known orthogonal…
We present in this paper some fundamental tools for developing matrix analysis over the complex quaternion algebra. As applications, we consider generalized inverses, eigenvalues and eigenvectors, similarity, determinants of complex…
In this work we describe an efficient implementation of a hierarchy of algorithms for the decomposition of dense matrices over the field with two elements (GF(2)). Matrix decomposition is an essential building block for solving dense…
Deep ReLU Networks can be decomposed into a collection of linear models, each defined in a region of a partition of the input space. This paper provides three results extending this theory. First, we extend this linear decompositions to…
Matrix decompositions are fundamental tools in the area of applied mathematics, statistical computing, and machine learning. In particular, low-rank matrix decompositions are vital, and widely used for data analysis, dimensionality…
We introduce the notion of a lowering-raising (or LR) triple of linear transformations on a nonzero finite-dimensional vector space. We show how to normalize an LR triple, and classify up to isomorphism the normalized LR triples. We…
We present a new formulation of the hyperbolic singular value decomposition (HSVD) for an arbitrary complex (or real) matrix without hyperexchange matrices and redundant invariant parameters. In our formulation, we use only the concept of…
Making new methods for quantum problems often relies on using basic operations in linear algebra. Often these routines are hidden behind well-known libraries that have been optimized over decades. Attempting to improve on those basic…
We give the images of the adjoint representations of exceptional simple Lie algebras by matrices over complex numbers. Next, we digitalize these matrices by the use of Maxima, which is a computer algebra system. These digitalized matrices…
Regular chains and triangular decompositions are fundamental and well-developed tools for describing the complex solutions of polynomial systems. This paper proposes adaptations of these tools focusing on solutions of the real analogue:…
In the present paper 2-local derivations on various algebras of infinite dimensional matrix-valued functions on a compact are considered. It is proved that every 2-local derivation on such algebra is a derivation. Also we explain that the…
The numerical solution of linear discrete ill-posed problems typically requires regularization, i.e., replacement of the available ill-conditioned problem by a nearby better conditioned one. The most popular regularization methods for…
Linear algebra's main concerns are sets of vectors, linear functions, subspaces, linear systems, matrices and concepts about those, such as whether the solution of linear system exists or is unique; a set of vectors is linearly independent…
We present a practical and efficient means to compute the singular value decomposition (svd) of a quaternion matrix A based on bidiagonalization of A to a real bidiagonal matrix B using quaternionic Householder transformations. Computation…
An arbitrary Mueller matrix can be decomposed into a sum of up to four deterministic Mueller-Jones matrices, with strengths given by the eigenvalues of an associated Hermitian matrix. A geometrical representation of the eigenvalues in terms…
A new algorithm to compute cylindrical algebraic decompositions (CADs) is presented, building on two recent advances. Firstly, the output is truth table invariant (a TTICAD) meaning given formulae have constant truth value on each cell of…
This article introduces the Hartwig-Spindelb\"{o}ck decomposition of dual complex matrices. We provide representations of some generalized inverses using this decomposition. Further, several characterizations are established for a complex…
We show that the discrete complex, and numerous hypercomplex, Fourier transforms defined and used so far by a number of researchers can be unified into a single framework based on a matrix exponential version of Euler's formula…
A novel factorization for the sum of two single-pair matrices is established as product of lower-triangular, tridiagonal, and upper-triangular matrices, leading to semi-closed-form formulas for tridiagonal matrix inversion. Subsequent…