Related papers: Supertropical matrix algebra III: Powers of matric…

The category $STROP_m$ of supertropical monoids, whose morphisms are transmissions, has the full--reflective subcategory $STROP$ of commutative semirings. In this setup, quotients are determined directly by equivalence relations, as ideals…

The map which takes a square matrix to its tropical eigenvalue-eigenvector pair is piecewise linear. We determine the cones of linearity of this map. They are simplicial but they do not form a fan. Motivated by statistical ranking, we also…

We solve a supersymmetric matrix model with a general potential. While matrix models usually describe surfaces, supersymmetry enforces a cancellation of bosonic and fermionic loops and only diagrams corresponding to so-called branched…

Some results in random matrices are generalized to supermatrices, in particular supermatrix integration is reduced to an integration over the eigenvalues and the resulting volume element is shown to be equivalent to a one dimensional…

We investigate spectral conditions on Hermitian matrices of roots of unity. Our main results are conjecturally sharp upper bounds on the number of residue classes of the characteristic polynomial of such matrices modulo ideals generated by…

Let k be an algebraically closed field of characteristic p \ge 0. We shall consider the problem of finding out a Jordan canonical form of J(\alpha,s) \otimes_{k} J(\beta,t), where J(\alpha,s) means the Jordan block with eigenvalue \alpha…

We introduce Z_3-graded objects which are the generalization of the more familiar Z_2-graded objects that are used in supersymmetric theories and in many models of non-commutative geometry. First, we introduce the Z_3-graded Grassmann…

The dimensions of sets of matrices of various types, with specified eigenvalue multiplicities, are determined. The dimensions of the sets of matrices with given Jordan form and with given singular value multiplicities are also found. Each…

This paper is devoted to the uniqueness problem of the power of a meromorphic function with its differential polynomial sharing a set. Our result will extend a number of results obtained in the theory of normal families. Some questions are…

In this paper we consider the decomposition of positive semidefinite matrices as a sum of rank one matrices. We introduce and investigate the properties of various measures of optimality of such decompositions. For some classes of positive…

We study elliptic surfaces over $\mathbb{Q}(T)$ with coefficients of a Weierstrass model being polynomials in $\mathbb{Q}[T]$ with degree at most 2. We derive an explicit expression for their rank over $\mathbb{Q}(T)$ depending on the…

Using matrix function theory, Perron-Frobenius theory, combinatorial matrix theory, and elementary number theory, we characterize, classify, and describe in terms of the Jordan canonical form the matrix pth-roots of imprimitive irreducible…

We compute correlation functions of inverse powers and ratios of characteristic polynomials for random matrix models with complex eigenvalues. Compact expressions are given in terms of orthogonal polynomials in the complex plane as well as…

We completely characterize the higher rank numerical range of the matrices of the form $J_n(\alpha)\oplus\beta I_m$, where $J_n(\alpha)$ is the $n\times n$ Jordan block with eigenvalue $\alpha$. Our characterization allows us to obtain…

We investigate traces of powers of random matrices whose distributions are invariant under rotations (with respect to the Hilbert--Schmidt inner product) within a real-linear subspace of the space of $n\times n$ matrices. The matrices we…

Let $p,q$ be coprime integers such that $|p|+|q|>2$. We characterize the matrices $A\in\mathcal{M}_n(\mathbb{C})$ such that $A^p$ and $A^q$ are similar. If $A$ is invertible, we prove that $A$ is a polynomial in $A^p$ and $A^q$. To achieve…

Even with the introduction of supercharacter theories, the representation theory of many unipotent groups remains mysterious. This paper constructs a family of supercharacter theories for normal pattern groups in a way that exhibit many of…

Tropicalization is a procedure for associating a polyhedral complex in Euclidean space to a subvariety of an algebraic torus. We study the question of which graphs arise from tropicalizing algebraic curves. By using Baker's specialization…

We define tensors, corresponding to cubic polynomials, which have the same exponent $\omega$ as the matrix multiplication tensor. In particular, we study the symmetrized matrix multiplication tensor $sM_n$ defined on an $n\times n$ matrix…

This paper presents a unified mathematical framework for inference in graphical models, building on the observation that graphical models are algebraic varieties. From this geometric viewpoint, observations generated from a model are…