Related papers: Matrix differentiation with diagrammatic notation
Differential reformulations of field theories are often used for explicit computations. We derive a one-matrix differential formulation of two-matrix models, with the help of which it is possible to diagonalize the one- and two-matrix…
We give a formula for matrix exponentials and partial fraction decompositions.
A natural consequence of the fractional calculus is its extension to a matrix order of differentiation and integration. A matrix-order derivative definition and a matrix-order integration arise from the generalization of the gamma function…
In this paper we derive a representation of an arbitrary real matrix M as the difference of a real matrix A and the transpose of its inverse. This expression may prove useful for progressing beyond known results for which the appearance of…
In this note, we define a Gaussian probability distribution over matrices. We prove some useful properties of this distribution, namely, the fact that marginalization, conditioning, and affine transformations preserve the matrix Gaussian…
Partitioning large matrices is an important problem in distributed linear algebra computing (used in ML among others). Briefly, our goal is to perform a sequence of matrix algebra operations in a distributed manner (whenever possible) on…
Computing derivatives of tensor expressions, also known as tensor calculus, is a fundamental task in machine learning. A key concern is the efficiency of evaluating the expressions and their derivatives that hinges on the representation of…
We introduce matrix and its block to the Dung's theory of argumentation frameworks. It is showed that each argumentation framework has a matrix representation, and the common extension-based semantics of argumentation framework can be…
A matrix approach to continuous iteration is proposed for general formal series. It leads, in particular, to an order{to{order iteration of the exponential function, and consequently to an algorithmic approach to tetration. Lower{order…
We review strategies for differentiating matrix-based computations, and derive symbolic and algorithmic update rules for differentiating expressions containing the Cholesky decomposition. We recommend new `blocked' algorithms, based on…
The matrices and their sub-blocks are introduced into the study of determining various extensions in the sense of Dung's theory of argumentation frameworks. It is showed that each argumentation framework has its matrix representations, and…
We present a new distributed representation in deep neural nets wherein the information is represented in native form as a matrix. This differs from current neural architectures that rely on vector representations. We consider matrices as…
Results of research of possibility of transformation of a difference equation into a system of the first-order difference equation are presented. In contrast to the method used previously, an unknown grid function is split into two new…
For a permutation $\pi$, and the corresponding permutation matrix, we introduce the notion of {\em discrete derivative}, obtained by taking differences of successive entries in $\pi$. We characterize the possible derivatives of…
Analytic perturbation theory for matrices and operators is an immensely useful mathematical technique. Most elementary introductions to this method have their background in the physics literature, and quantum mechanics in particular. In…
In an earlier work, the author together with Guo [Hermitian adjacency matrix of digraphs and mixed graphs, J. Graph Theory 85 (2017) 217-248] introduced the Hermitian adjacency matrix of directed (and partially directed) graphs. However, it…
In finance, economics and many other fields, observations in a matrix form are often observed over time. For example, many economic indicators are obtained in different countries over time. Various financial characteristics of many…
A new matrix operation based on inserting columns and rows, similarly to the mediant operation between fractions, gives rise to the Farey determinants matrix or, equivalently, the matrix of the numerators of the differences of Farey…
This course, intended for undergraduates familiar with elementary calculus and linear algebra, introduces the extension of differential calculus to functions on more general vector spaces, such as functions that take as input a matrix and…
The purpose of this note is to present a formulation of a given nonlinear ordinary differential equation into an equivalent system of linear ordinary differential equations. It is evident that the easiness of a such procedure would be able…