Related papers: The square root theorem for positive semidefinite …
Positive semidefinite matrices partitioned into a small number of Hermitian blocks have a remarkable property. Such a matrix may be written in a simple way from the sum of its diagonal blocks
For a 4th order 3-dimensional cyclic symmetric tensor, a sufficient and necessary condition is bulit for its positive semi-definiteness. A sufficient and necessary condition of positive definiteness is showed for a 4th order $n$-dimensional…
We discuss the possibility to represent smooth nonnegative matrix-valued functions as finite linear combinations of fixed matrices with positive real-valued coefficients whose square roots are Lipschitz continuous. This issue is reduced to…
In the paper, by establishing the monotonicity of some functions involving the sine and cosine functions, the authors provide concise proofs of some known inequalities and find some new sharp inequalities involving the Seiffert,…
In recent years several classes of structured matrices are extended to classes of tensors in the context of tensor complementarity problem. The tensor complementarity problem is a class of nonlinear complementarity problem where the…
We give an elementary proof of a Caratheodory-type result on the invertibility of a sum of matrices, due first to Facchini and Barioli. The proof yields a polynomial identity, expressing the determinant of a large sum of matrices in terms…
A synaptic algebra is a common generalization of several ordered algebraic structures based on algebras of self-adjoint operators, including the self-adjoint part of an AW*-algebra. In this paper we prove that a synaptic algebra A has the…
In this paper we establish some applications of the Scherer-Hol's theorem for polynomial matrices. Firstly, we give a representation for polynomial matrices positive definite on subsets of compact polyhedra. Then we establish a…
A real symmetric matrix $M$ is completely positive semidefinite if it admits a Gram representation by (Hermitian) positive semidefinite matrices of any size $d$. The smallest such $d$ is called the (complex) completely positive semidefinite…
A general theorem on factorization of matrices with polynomial entries is proven and it is used to reduce polynomial Darboux matrices to linear ones. Some new examples of linear Darboux matrices are discussed.
I provide an alternate and self-contained proof of Botha's theorem on products of idempotent and square-zero matrices where the product contains two square-zero factors, and provide a conclusive characterisation of products of singular…
We give several descriptions of positive quadrature formulas which are exact for trigonometric -, respectively, Laurent polynomials of degree less or equal $n-1-m$, $0\leq m\leq n-1$. A complete and simple description is obtained with the…
We investigate the representation of symmetric polynomials as a sum of squares. Since this task is solved using semidefinite programming tools we explore the geometric, algebraic, and computational implications of the presence of discrete…
We consider a symmetric matrix, the entries of which depend linearly on some parameters. The domains of the parameters are compact real intervals. We investigate the problem of checking whether for each (or some) setting of the parameters,…
Positive definite forms $f$ which are sums of squares are constructed to have the additional property that the members of any collection of forms whose squares sum to $f$ must share a nontrivial complex root.
In this note we study some sequences whose ratio converges to the square root of rationals. Further we analyze some related sequences obtained when the above mentioned ratio simplifies.
This article provides a simple proof of the quadratic formula, which also produces an efficient and natural method for solving general quadratic equations. The derivation is computationally light and conceptually natural, and has the…
We prove several formulas for the distribution of positive roots.
Let $H$ be a positive semi-definite matrix partitioned in $\beta\times \beta$ Hermitian blocks, $H=[A_{s,t}]$, $1\le s,t,\le \beta$. Then, for all symmetric norms, {equation*} \| H \| \le \| \sum_{s=1}^{\beta} A_{s,s} \|. {equation*} The…
This brief note concerns the invertibility of certain alternant matrices. In particular those that consisting of polynomials and products of polynomials and logarithms are shown to be invertible under appropriate conditions on the degrees…