Related papers: Sarrus' Quilt: A Novel Way to Compute Determinants
We evaluate the determinant of a matrix whose entries are elliptic hypergeometric terms and whose form is reminiscent of Sylvester matrices. A hypergeometric determinant evaluation of a matrix of this type has appeared in the context of…
The use of quadratic residues to construct matrices with specific determinant values is a familiar problem with connections to many areas of mathematics and statistics. Our research has focused on using cubic residues to construct matrices…
Matrices can be augmented by adding additional columns such that a partitioning of the matrix in blocks of rows defines mutually orthogonal subspaces. This augmented system can then be solved efficiently by a sum of projections onto these…
In order to compute the log-likelihood for high dimensional spatial Gaussian models, it is necessary to compute the determinant of the large, sparse, symmetric positive definite precision matrix, Q. Traditional methods for evaluating the…
Motivated by complexity questions in integer programming, this paper aims to contribute to the understanding of combinatorial properties of integer matrices of row rank $r$ and with bounded subdeterminants. In particular, we study the…
In this note we introduce a determinant and then give its evaluating formula. The determinant turns out to be a generalization of the well-known ballot and Fuss-Catalan numbers, which is believed to be new. The evaluating formula is proved…
This note gives a simple analysis of a randomized approximation scheme for matrix multiplication proposed by Sarlos (2006) based on a random rotation followed by uniform column sampling. The result follows from a matrix version of…
The theory of two-dimensional linear quaternion-valued differential equations (QDEs) was recently established (see Kou and Xia, SAPM). Some profound differences between QDEs and ODEs were observed. Also, an algorithm to evaluate the…
Without adhering to any specific model, we have presented 4 X 4 quark mixing matrix as an extension of the 3 X 3 PDG parametrization of the CKM matrix. Using unitarity constraints as well as the hierarchy among the elements of the 3 X 3 CKM…
This note deals with two topics of linear algebra. We give a simple and short proof of the multiplicative property of the determinant and provide a constructive formula for rotations. The derivation of the rotation matrix relies on simple…
In this paper, we present a new formula of the determinant tensor $det_n$ for $n \times n$ matrices. In \cite{kim2023newdet4}, Kim, Ju, and Kim found a new formula of $4 \times 4$ determinant tensor $det_4$ which is available when the base…
An algorithm to compute efficiently the first two derivatives of (very) large multideterminant wavefunctions for quantum Monte Carlo calculations is presented. The calculation of determinants and their derivatives is performed using the…
We investigate determinants of random unitary pencils (with scalar or matrix coefficients), which generalize the characteristic polynomial of a single unitary matrix. In particular we examine moments of such determinants, obtained by…
The determinant of an $N \times N$ circulant matrix $M = {\rm CIRC}[x_0, x_1, ..., x_{N-1}$] can be expanded in the form det$ ~M= \sum C_{a_0 a_1 ...a_{N-1}} x_{a_0} x_{a_1}...x_{a_{N-1}}$. By using the generating function of a restricted,…
There is a digraph corresponding to every square matrix over $\mathbb{C}$. We generate a recurrence relation using the Laplace expansion to calculate the characteristic, and permanent polynomials of a square matrix. Solving this recurrence…
We consider evaluation of matrix elements with the coupled-cluster method. Such calculations formally involve infinite number of terms and we devise a method of partial summation (dressing) of the resulting series. Our formalism is built…
Algorithms involving Gaussian processes or determinantal point processes typically require computing the determinant of a kernel matrix. Frequently, the latter is computed from the Cholesky decomposition, an algorithm of cubic complexity in…
In an earlier version of this paper \cite{6} a formula for the 4th cumulant of the Rosenblatt distribution was derived. In this paper formulas for the 3rd, 4th and 5th cumulant are derived using two methods. In the first method the…
Covariance matrix estimation is a fundamental statistical task in many applications, but the sample covariance matrix is sub-optimal when the sample size is comparable to or less than the number of features. Such high-dimensional settings…
This paper considers the problem of estimating a change point in the covariance matrix in a sequence of high-dimensional vectors, where the dimension is substantially larger than the sample size. A two-stage approach is proposed to…