相关论文: Evaluating Two Determinants
We compute two parametric determinants in which rows and columns are indexed by compositions, where in one determinant the entries are products of binomial coefficients, while in the other the entries are products of powers. These results…
This is a complement to my previous article "Advanced Determinant Calculus" (S\'eminaire Lotharingien Combin. 42 (1999), Article B42q, 67 pp.). In the present article, I share with the reader my experience of applying the methods described…
A new class of structured matrices is presented and a closed form formula for their determinant is established. This formula has strong connections with the one for Vandermonde matrices.
We prove several evaluations of determinants of matrices, the entries of which are given by the recurrence $a_{i,j}=a_{i-1,j}+a_{i,j-1}$, or variations thereof. These evaluations were either conjectured or extend conjectures by Roland…
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…
Pausinger recently investigated a special determinant involving prime numbers. In this short note we point out that this type of determinants was already known in linear algebra and its computation is unrelated to prime numbers.
This paper presents a method for expressing the determinant of an N {\times} N complex block matrix in terms of its constituent blocks. The result allows one to reduce the determinant of a matrix with N^2 blocks to the product of the…
In this (mostly expository) paper I want to share some observations prompted by a class of matrices whose determinants are Catalan numbers. Considering different methods of proof we obtain some generalizations and q-analogues and…
The purpose of this article is threefold. First, it provides the reader with a few useful and efficient tools which should enable her/him to evaluate nontrivial determinants for the case such a determinant should appear in her/his research.…
Two known computation methods and one new computation method for matrix determinant over an integral domain are discussed. For each of the methods we evaluate the computation times for different rings and show that the new method is the…
In his work on the twenty vertex model, Di Francesco [Electron. J. Combin. 28(4) (2021), Paper No. 4.38] found a determinant formula for the number of configurations in a specific such model, and he conjectured a closed form product formula…
A formula is presented for the determinant of the second additive compound of a square matrix in terms of coefficients of its characteristic polynomial. This formula can be used to make claims about the eigenvalues of polynomial matrices,…
We evaluate four families of determinants of matrices, where the entries are sums or differences of generating functions for paths consisting of up-steps, down-steps and level steps. By specialisation, these determinant evaluations have…
We study determinants of matrices whose entries are powers of Fibonacci numbers. We then extend the results to include entries that are powers of generalized Fibonacci numbers defined as a second-order linear recurrence relation. These…
We prove two inequalities regarding the ratio $\det(A+D)/\det A$ of the determinant of a positive-definite matrix $A$ and the determinant of its perturbation $A+D$. In the first problem, we study the perturbations that happen when positive…
The completeness of the group classification of systems of two linear second-order ordinary differential equations with constant coefficients is delineated in the paper. The new cases extend what has been done in the literature. These cases…
We present here a new method for evaluating determinants -- the reduction method. Firstly, in the section 2, we apply it to third-order determinants and after, in the section 3, we generalize it to higher-order determinants. In the section…
We prove the determinant connectivity matrix formula. Mathematically, the proof introduces novel techniques based on an algebraic approach and connectivity properties. Although this is the second part of a previous paper and has its…
We give a new combinatorial explanation for well-known relations between determinants and traces of matrix powers. Such relations can be used to obtain polynomial-time and poly-logarithmic space algorithms for the determinant. Our new…
We evaluate determinants of "spiral" matrices, which are matrices in which entries are spiralling from the centre of the matrices towards the outside, with prescribed increments from one entry to the next depending on whether one moves…