Related papers: Determinantal inequalities for block triangular ma…

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…

Let $G$ be an abelian group and $\mathbb{K}$ an algebraically closed field of characteristic zero. A. Valenti and M. Zaicev described the $G$-gradings on upper block-triangular matrix algebras provided that $G$ is finite. We prove that…

Let $X$ be a genus zero compact polyhedral surface (the Riemann sphere equipped with a flat conical metric $m$). We derive the variational formulas for the determinant of the Laplacian, ${\rm det}\,\Delta^m$, on $X$ under infinitesimal…

In this paper, we present some extensions of the Young and Heinz inequalities for the Hilbert-Schmidt norm as well as any unitarily invariant norm. Furthermore, we give some inequalities dealing with matrices. More precisely, for two…

For an irreducible orthogonal character $\chi $ of even degree there is a unique square class $\det({\chi })$ in the character field such that the invariant quadratic forms in any $L$-representation affording $\chi $ have determinant in…

We determine the irreducible components of the space of 3x3 matrices of linear forms with vanishing determinant. We show that there are four irreducible components and we identify them concretely. In particular, under elementary row and…

For positive semidefinite $n\times n$ matrices $A$ and $B$, the singular value inequality $(2+t)s_{j}(A^{r}B^{2-r}+A^{2-r}B^{r})\leq 2s_{j}(A^{2}+tAB+B^{2})$ is shown to hold for $r=\frac{1}{2}, 1, \frac{3}{2}$ and all $-2<t\leq 2$.

Recently there has been several works estimating the number of $n\times n$ matrices with elements from some finite sets $\mathcal X$ of arithmetic interest and of a given determinant. Typically such results are compared with the trivial…

Let F be a field. We investigate the greatest possible dimension t_n(F) for a vector space of n-by-n matrices with entries in F and in which every element is triangularizable over the ground field F. It is obvious that t_n(F) is greater…

The paper is devoted to complete proofs of theorems on consistency on cubic lattices for $3 \times 3$ determinants. The discrete nonlinear equations on $\mathbb{Z}^2$ defined by the condition that the determinants of all $3 \times 3$…

One important question in algebraic complexity is understanding the complexity of polynomial ideals (Grochow, Bulletin of EATCS 131, 2020). Andrews and Forbes (STOC 2022) studied the determinantal ideals $I^{\det}_{n,m,r}$ generated by the…

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,…

We determine the probability that a random n x n symmetric matrix over {1, 2, ... , m} has determinant divisible by m.

In this paper we compute Gr\"obner bases for determinantal ideals of the form $I_{1}(XY)$, where $X$ and $Y$ are both matrices whose entries are indeterminates over a field $K$. We use the Gr\"obner basis structure to determine Betti…

The paper develops elementary linear algebra methods to compute the determinants of the tensor symmetrizations of quadratic and hermitian forms over fields of good characteristic. Explicit results are given for the partitions $(n)$,…

A subspace of the space, L(n), of traceless complex $n\times n$ matrices can be specified by requiring that the entries at some positions $(i,j)$ be zero. The set, $I$, of these positions is a (zero) pattern and the corresponding subspace…

Let y1, y2, y3, a1, a2, a3 > 0 be such that y1 y2 y3 = a1 a2 a3 and y1 + y2 + y3 >= a1 + a2 + a3, y1 y2 + y2 y3 + y1 y3 >= a1 a2 + a2 a3 + a1 a3. Then the following inequality holds (log y1)^2 + (log y2)^2 + (log y3)^2 >= (log a1)^2 + (log…

We consider the set $\mathcal M_n(\mathbb Z; H)$ of $n\times n$-matrices with integer elements of size at most $H$ and obtain upper bounds on the number of matrices from $\mathcal M_n(\mathbb Z; H)$, for which the characteristic polynomial…

Let $G$ be a graph(directed or undirected) having $k$ number of blocks. A $\mathcal{B}$-partition of $G$ is a partition into $k$ vertex-disjoint subgraph $(\hat{B_1},\hat{B_1},\hdots,\hat{B_k})$ such that $\hat{B}_i$ is induced subgraph of…

The notion of isomorphism of stable AF-C*-algebras is considered in this paper in the case when the corresponding Bratteli diagram is stationary, i.e., is associated with a single square primitive nonsingular incidence matrix.…