相关论文: Determinantal inequalities for block triangular ma…

In this paper we consider pentadiagonal $(n+1)\times(n+1)$ matrices with two subdiagonals and two superdiagonals at distances $k$ and $2k$ from the main diagonal where $1\le k<2k\le n$. We give an explicit formula for their determinants and…

A. Mukhopadhyay, M. R. Murty and K. Srinivas (http://arxiv.org/abs/0808.0418) have recently studied various arithmetic properties of the discriminant $\Delta_n(a,b)$ of the trinomial $f_{n,a,b}(t) = t^n + at + b$, where $n \ge 5$ is a fixed…

The characteristic polynomial of an $r$-tuple $(A_1,..., A_r)$ of $n \times n$ matrices is the determinant $\det(x_0 I + x_1 A_1 + ... + x_r A_r)$. We show that if $r$ is at least 3 and $A = (A_1,..., A_r)$ is an $r$-tuple of matrices in…

Consider rectangular matrices over a local ring R. In the previous work we have obtained criteria for block-diagonalization of such matrices, i.e. U A V=A_1\oplus A_2, where U,V are invertible matrices over R. In this short note we extend…

In this work, the determinants of matrices constructed by evaluating homogeneous bivariate polynomials at pairs of vectors are investigated. For a polynomial $p(x,y)=\sum\limits_{i=0}^k \alpha_i x^{k-i}y^i$, an explicit factorization of the…

We define three types of upper (and lower) triangular blocked tensors, which are all generalizations of the triangular blocked matrices. We study some basic properties and characterizations of these three types of triangular blocked…

A square matrix is $k$-Toeplitz if its diagonals are periodic sequences of period $k$. We find universal formulas for the determinant, the characteristic polynomial, some eigenvectors, and the entries of the inverse of any tridiagonal…

Moment inequality for quadratic forms of random vectors is of particular interest in covariance matrix testing and estimation problems. In this paper, we prove a Rosenthal-type inequality, which exhibits new features and certain improvement…

Given a matrix $A$, let $A_{I,J}$ denote the submatrix of $A$ determined by rows $I$ and columns $J$. Fischer's Inequalities state that for each $n \times n$ Hermitian positive semidefinite matrix $A$, and each subset $I$ of…

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…

We first give an Oppenheim type determinantal inequality for the Khatri-Rao product of two block positive semidefinite matrices, and then we extend our result to multiple block matrices. As products, the extensions of Oppenheim type…

We first present a determinant inequality related to partial traces for positive semidefinite block matrices. Our result extends a result of Lin [Czech. Math. J. 66 (2016)] and improves a result of Kuai [Linear Multilinear Algebra 66…

Let T be an n by n zero-square matrix over a commutative unital ring R. We show that T is similar to a multiple of E_1n if R is a GCD domain and n = 2, if R is a GCD domain with 2 not zero divisor and n = 3, but there are matrices which are…

The main question we raise here is the following one: given a real orthogonal n by n matrix X, is it true that there exists a rational orthogonal matrix Y having the same zero-pattern? We conjecture that this is the case and prove it for…

Yuan and Leng (2007) gave a generalization of Ky Fan's determinantal inequality, which is a celebrated refinement of the fundamental Brunn-Minkowski inequality $(\det (A+B))^{1/n} \ge (\det A)^{1/n} +(\det B)^{1/n}$, where $A$ and $B$ are…

Due to their rich algebraic structures and various applications, circulant matrices have been of interest and continuously studied. In this paper, the notions of Binomial-related matrices have been introduced. Such matrices are circulant…

The Drazin index is a fundamental invariant in the analysis of singular matrices and their generalized inverses. While sharp results are available for block triangular matrices, the corresponding theory for anti-triangular block matrices is…

Let $A_i$ and $B_i$ be positive definite matrices for every $i=1,\cdots,m.$ Let $Z=[Z_{ij}]$ be the block matrix, where $Z_{ij}=B_i^{^\frac{1}{_2}}\left(\displaystyle\sum_{k=1}^mA_k\right)B_j^{^\frac{1}{_2}}$ for every $ i,j=~1,\cdots,m$.…

The exponential of the triangular matrix whose entries in the diagonal at distance $n$ from the principal diagonal are all equal to the sum of the inverse of the divisors of $n$ is the triangular matrix whose entries in the diagonal at…

We study higher order determinantal varieties obtained by considering generic $m\times n$ ($m \le n$) matrices over rings of the form $F[t]/(t^k)$, and for some fixed $r$, setting the coefficients of powers of $t$ of all $r \times r$ minors…