Related papers: On sufficient conditions for the total positivity …
A real sequence $(a_k)_{k=0}^\infty$ is called {\it totally positive} if all minors of the infinite Toeplitz matrix $ \left\| a_{j-i} \right\|_{i, j =0}^\infty$ are nonnegative (here $a_k=0$ for $k<0$). In this paper, which continues our…
Using the Perron-Frobenius eigenfunction and eigenvalue, each finite irreducible nonnegative matrix $A$ can be transformed into a probability kernel $P$. This was generalized by David Vere-Jones who gave necessary and sufficient conditions…
We prove that the only entrywise transforms of rectangular matrices which preserve total positivity or total non-negativity are either constant or linear. This follows from an extended classification of preservers of these two properties…
A real sequence $(b_k)_{k=0}^\infty$ is called totally positive if all minors of the infinite matrix $ \left\| b_{j-i} \right\|_{i, j =0}^\infty$ are nonnegative (here $b_k=0$ for $k<0$). In this paper, we investigate the problem of…
It is known that for a totally positive (TP) matrix, the eigenvalues are positive and distinct and the eigenvector associated with the smallest eigenvalue is totally nonzero and has an alternating sign pattern. Here, a certain weakening of…
We give a short proof of a recent result of Drury on the positivity of a $3\times 3$ matrix of the form $(\|R_i^*R_j\|_{\rm tr})_{1 \le i, j \le 3}$ for any rectangular complex (or real) matrices $R_1, R_2, R_3$ so that the multiplication…
A square matrix $M$ with real entries is said to be algebraically positive (AP) if there exists a real polynomial $p$ such that all entries of the matrix $p(M)>0$. A square sign pattern matrix $S$ is said to allow algebraic positivity if…
If $(X,J)$ is an almost complex manifold, then a function $u$ is said to be plurisubharmonic on $X$ if it is upper semi-continuous and its restriction to every local pseudo-holomorphic curve is subharmonic. As in the complex case, it is…
We propose a numerical method, based on the shift-and-invert power iteration, that answers whether a symmetric matrix is positive definite ("yes") or not ("no"). Our method uses randomization. But, it returns the correct answer with high…
In order to compute the Schmidt decomposition of $A\in M_k\otimes M_m$, we must consider an associated self-adjoint map. Here, we show that if $A$ is positive under partial transposition (PPT) or symmetric with positive coefficients (SPC)…
Log-concavity and almost log-convexity of the cycle index polynomials were proved by Bender and Canfield [J. Combin. Theory Ser. A 74 (1996)]. Schirmacher [J. Combin. Theory Ser. A 85 (1999)] extended them to $q$-log-concavity and almost…
A linear map between matrix spaces is positive if it maps positive semidefinite matrices to positive semidefinite ones, and is called completely positive if all its ampliations are positive. In this article quantitative bounds on the…
We prove that if $k$ is a positive integer then for every finite field $\mathbb{F}$ of cardinality $q\neq 2$ and for every positive integer $n$ such that $q^n>(k-1)^4$, every $n\times n$ matrix over $\mathbb{F}$ can be expressed as a sum of…
We show that the maximal number of equal entries in a totally positive (resp. totally nonsingular) $n\textrm{-by-}n$ matrix is $\Theta(n^{4/3})$ (resp. $\Theta(n^{3/2}$)). Relationships with point-line incidences in the plane, Bruhat order…
For two sets $A$ and $M$ of positive integers and for a positive integer $n$, let $p(n,A,M)$ denote the number of partitions of $n$ with parts in $A$ and multiplicities in $M$, that is, the number of representations of $n$ in the form…
We present an extension of the Hardy--Littlewood inequality for multilinear forms. More precisely, let $\mathbb{K}$ be the real or complex scalar field and $m,k$ be positive integers with $m\geq k\,$ and $n_{1},\dots ,n_{k}$ be positive…
The composition operators preserving total non-negativity and total positivity for various classes of kernels are classified, following three themes. Letting a function act by post composition on kernels with arbitrary domains, it is shown…
A proof using the theory of completely positive maps is given to the fact that if $A \in M_2$, or $A \in M_3$ has a reducing eigenvalue, then every bounded linear operator $B$ with $W(B) \subseteq W(A)$ has a dilation of the form $I \otimes…
Let $G_{m \times n}$ be an $m \times n$ real random matrix whose elements are independent and identically distributed standard normal random variables, and let $\kappa_2(G_{m \times n})$ be the 2-norm condition number of $G_{m \times n}$.…
A matrix is totally positive if all of its minors are positive. This notion of positivity coincides with the type A version of Lusztig's more general total positivity in reductive real-split algebraic groups. Since skew-symmetric matrices…