Matrices autosimilaires

对偶四元数埃尔米特矩阵的Moore行列式

Moore Determinant of Dual Quaternion Hermitian Matrices

该研究将四元数埃尔米特矩阵的Chen、Moore行列式推广到对偶四元数埃尔米特矩阵，证明两类行列式相等且等于特征值乘积，还研究了其特征多项式。

Determinants of some Special Matrices over Commutative Finite Chain Rings

Circulant matrices over finite fields and over commutative finite chain rings have been of interest due to their nice algebraic structures and wide applications. In many cases, such matrices over rings have a closed connection with diagonal matrices over their extension rings. In this paper, the determinants of diagonal and circulant matrices over commutative finite chain rings $R$ with residue field $\mathbb{F}_q$ are studied. The number of $n\times n$ diagonal matrices over ${R}$ of determinant $a$ is determined for all elements $a$ in $ {R}$ and for all positive integers $n$. Subsequently, …

Determinants of Matrices over Commutative Finite Principal Ideal Rings

In this paper, the determinants of $n\times n$ matrices over commutative finite chain rings and over commutative finite principal ideal rings are studied. The number of $n\times n$ matrices over a commutative finite chain ring ${R}$ of a fixed determinant $a$ is determined for all $a\in {R}$ and positive integers $n$. Using the fact that every commutative finite principal ideal ring is a product of commutative finite chain rings, the number of $n\times n$ matrices of a fixed determinant over a commutative finite principal ideal ring is shown to be multiplicative, and hence, it can be determine…

方阵的行列式

Déterminant d'une matrice carrée

本文研究可逆矩阵的p阶子式矩阵，并针对SL(n)与SO(n)中的矩阵分析了特殊情形。

Some matrices with nilpotent entries, and their determinants

We study algebraic properties of matrices whose rows are mutual neighbours, and are also neigbours of 0 ("neighbour" in the sense of a certain nilpotency condition). The intended application is in synthetic differential geometry. For a square matrix of this kind, the product of the diagonal entries equals the determinant, modulo a factor n!

Determinants in the Kronecker product of matrices: The incidence matrix of a complete graph

We investigate the least common multiple of all subdeterminants, lcmd(A x B), of a Kronecker product of matrices, of which one is an integral matrix A with two columns and the other is the incidence matrix of a complete graph with n vertices. We prove that this quantity is the least common multiple of lcmd(A) to the power n-1 and certain binomial functions of the entries of A.

对偶四元数矩阵的拟行列式与右特征值

Quasi-determinant and right eigenvalues of dual quaternion matrices

本文研究对偶四元数矩阵的拟行列式与右特征值，证明特征值必为拟特征多项式的根，并给出Hermitian矩阵拟行列式与特征值模平方乘积的等价关系。

Correspondence between Row-Column Determinants and Quasideterminants of Matrices over Quaternion Algebra

In this paper, we considered the theory of quasideterminants and row and column determinants. We considered the application of this theory to the solving of a system of linear equations in quaternion algebra. We established correspondence between row and column determinants and quasideterminants of matrix over quaternion algebra.

Determinants for nxn matrices and the symmetric Newton formula in the 3x3 case

One of the aims of this paper is to provide a short survey on the Z2-graded, the symmetric and the left (right) generalizations of the classical determinant theory for square matrices with entries in an arbitrary (possibly non-commutative) ring. This will put us in a position to give a motivation for our main results. We use the preadjoint matrix to exhibit a general trace expression for the symmetric determinant. The symmetric version of the classical Newton trace formula is also presented in the 3x3 case.

Determinants preserving maps on the spaces of symmetric matrices and skew-symmetric matrices

Denote $Σ_n$ and $Q_n$ the set of all $n \times n$ symmetric and skew-symmetric matrices over a field $\mathbb{F}$, respectively, where $char(\mathbb{F})\neq 2$ and $\lvert \mathbb{F} \rvert \geq n^2+1$. A characterization of $ϕ,ψ:Σ_n \rightarrow Σ_n$, for which at least one of them is surjective, satisfying $$\det(ϕ(x)+ψ(y))=\det(x+y)\qquad(x,y\in Σ_n)$$ is given. Furthermore, if $n$ is even and $ϕ,ψ:Q_n \rightarrow Q_n$, for which $ψ$ is surjective and $ψ(0)=0$, satisfy $$\det(ϕ(x)+ψ(y))=\det(x+y)\qquad(x,y\in Q_n),$$ then $ϕ=ψ$ and $ψ$ must be a bijective linear map preserving the determina…

关于交换有限链环上具有指定行列式的对称三对角矩阵的计数

On the Enumeration of Symmetric Tridiagonal Matrices with prescribed Determinant over Commutative Finite Chain Rings

本文研究有限域和交换有限链环上具有指定行列式的对称三对角矩阵计数问题，利用行列式递推、二次特征和分层约化方法，给出了统一计数框架及显式公式。

Generalizations of the Maillet Determinant

We consider several extensions of the Maillet determinant studied by Malo, Turnbull, and Carlitz and Olson, and derive properties of the underlying matrices. In particular, we compute the eigenvectors and eigenvalues of these matrices, which yield formulas for these new determinants.

Trace and determinant preserving maps of matrices

Suppose a map $ϕ$ on the set of positive definite matrices satisfies $\det(A+B)=\det(ϕ(A)+ϕ(B))$. Then we have $${\rm tr}(AB^{-1}) = {\rm tr}(ϕ(A){ϕ(B)}^{-1}).$$ Through this viewpoint, we show that $ϕ$ is of the form $ϕ(A)= M^*AM$ or $ϕ(A)= M^*A^tM$ for some invertible matrix $M$ with $\det (M^*M)=1$. We also characterize the map $ϕ: \mathcal{S} \rightarrow \mathcal{S}$ preserving the determinant of convex combinations in $\mathcal{S}$ by using similar method. Here $\mathcal{S}$ can be the set of complex matrices, positive definite matrices, symmetric matrices, and upper triangular matrices.

Partial determinants of Kronecker products

Let $\det_2(A)$ be the block-wise determinant (partial determinant). We consider the condition for completing the determinant $\det(\det_2(A)) = \det(A),$ and characterize the case for an arbitrary Kronecker product $A$ of matrices over an arbitrary field. Further insisting that $\det_2(AB)=\det_2(A)\det_2(B)$, for Kronecker products $A$ and $B$, yields a multiplicative monoid of matrices. This leads to a determinant-root operation $\text{Det}$ which satisfies $\text{Det}(\text{Det}_2(A)) = \text{Det}(A)$ when $A$ is a Kronecker product of matrices for which $\text{Det}$ is defined.

计算分块矩阵的行列式

Calculating Determinants of Block Matrices

提出一种将N×N复分块矩阵的行列式表示为各分块组合的行列式乘积的方法，用于简化多离散变量物理系统的解析计算。

Determinants of Block Matrices with Noncommuting Blocks

Let $M$ be an $mn\times mn$ matrix over a commutative ring $R$. Divide $M$ into $m \times m$ blocks. Assume that the blocks commute pairwise. Consider the following two procedures: (1) Evaluate the $n \times n$ determinant formula at these blocks to obtain an $m \times m$ matrix, and take the determinant again to obtain an element of $R$; (2) Take the $mn \times mn$ determinant of $M$. It is known that the two procedures give the same element of $R$. We prove that if only certain pairs of blocks of $M$ commute, then the two procedures still give the same element of $R$, for a suitable definiti…

Generalized Pascal Triangles and Toeplitz Matrices

The purpose of this article is to study determinants of matrices which are known as generalized Pascal triangles (see [1]). We present a factorization by expressing such a matrix as a product of a unipotent lower triangular matrix, a Toeplitz matrix and a unipotent upper triangular matrix. The determinant of a generalized Pascal matrix equals thus the determinant of a Toeplitz matrix. This equality allows us to evaluate a few determinants of generalized Pascal matrices associated to certain sequences. In particular, we obtain families of quasi-Pascal matrices whose principal minors generate an…

Properties of determinantal polynomials of subspaces of matrices over a finite field

Let K be a field and let M_n(K) denote the space of n x n matrices with entries in K. Let M be a subspace of M_n(K) of dimension d with the property that there are elements in M with non-zero determinant. Given a basis of M, we define the determinantal polynomial P_{M} of M with respect to the basis. It is a homogeneous polynomial of degree n in d indeterminates that gives the determinant of any element of M by evaluation in K^d. This paper investigates the interrelationship of M and P_{M}. We confine ourselves to finite fields K, where we can obtain useful information by applying the Lang-Wei…

计算Toeplitz矩阵行列式的一种快速初等算法

A fast elementary algorithm for computing the determinant of toeplitz matrices

提出一种基于伴随矩阵幂的初等证明方法，实现Toeplitz矩阵行列式的快速计算，复杂度为O(k^2 log n + k^3)。