It is shown that for positive real numbers $ 0<λ_{1}<\dots<λ_{n}$, $\left[\frac{1}{β({λ_i}, {λ_j})}\right]$, where $ β(\cdot,\cdot)$ denotes the beta function, is infinitely divisible and totally positive. For $ \left[\frac{1}{β({i},{j})}\right]$, the Cholesky decomposition and successive elementary bidiagonal decomposition are computed. Let $\mathfrak w(n)$ be the $n$th Bell number. It is proved that $\left[\mathfrak w(i+j)\right]$ is a totally positive matrix but is infinitely divisible only upto order $4$. It is also shown that the symmetrized Stirling matrices are totally positive.
In their recent paper "The spectral norm of a Horadam circulant matrix", Merikoski, Haukkanen, Mattila and Tossavainen study under which conditions the spectral norm of a general real circulant matrix ${\bf C}$ equals the modulus of its row/column sum. We improve on their sufficient condition until we have a necessary one. Our results connect the above problem to positivity of sufficiently high powers of the matrix ${\bf C^\top C}$. We then generalize the result to complex circulant matrices.
We review some of the significant generalizations and applications of the celebrated Douglas theorem on the equivalence of factorization, range inclusion, and majorization of operators. We then apply it to find a characterization of the positivity of $2\times 2$ block matrices of operators in Hilbert spaces and finally describe the nature of such block matrices and provide several ways for showing their positivity.
The matrix $S = [1+x_i y_j]_{i,j=1}^{n}, 0<x_1<\cdots<x_n,\, 0<y_1<\cdots<y_n$, has gained importance lately due to its role in powers preserving total nonnegativity. We give an explicit decomposition of $S$ in terms of elementary bidiagonal matrices, which is analogous to the Neville decomposition. We give a bidiagonal decomposition of $S^{\circ m}=[(1+x_iy_j)^m]$ for positive integers $1\leq m \leq n-1$. We also explore the total positivity of Hadamard powers of another important class of matrices called mean matrices.
A classical theorem proved in 1942 by I.J. Schoenberg describes all real-valued functions that preserve positivity when applied entrywise to positive semidefinite matrices of arbitrary size; such functions are necessarily analytic with non-negative Taylor coefficients. Despite the great deal of interest generated by this theorem, a characterization of functions preserving positivity for matrices of fixed dimension is not known. In this paper, we provide a complete description of polynomials of degree $N$ that preserve positivity when applied entrywise to matrices of dimension $N$. This is the …
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 $R_i^*R_j$ is compatible for all $i, j$, where $\|\cdot\|_{\rm tr}$ denotes the trace norm. We then give a complete analysis of the problem when the trace norm is replaced by other unitarily invariant norms.
Functions preserving Loewner positivity when applied entrywise to positive semidefinite matrices have been widely studied in the literature. Following the work of Schoenberg [Duke Math. J. 9], Rudin [Duke Math. J. 26], and others, it is well-known that functions preserving positivity for matrices of all dimensions are absolutely monotonic (i.e., analytic with nonnegative Taylor coefficients). In this paper, we study functions preserving positivity when applied entrywise to sparse matrices, with zeros encoded by a graph $G$ or a family of graphs $G_n$. Our results generalize Schoenberg and Rudi…
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 for matrices of fixed dimension. We also prove that the same assertions hold upon working only with symmetric matrices; for total-positivity preservers our proofs proceed through solving two totally positive completion problems.
A classical result by Schoenberg (1942) identifies all real-valued functions that preserve positive semidefiniteness (psd) when applied entrywise to matrices of arbitrary dimension. Schoenberg's work has continued to attract significant interest, including renewed recent attention due to applications in high-dimensional statistics. However, despite a great deal of effort in the area, an effective characterization of entrywise functions preserving positivity in a fixed dimension remains elusive to date. As a first step, we characterize new classes of polynomials preserving positivity in fixed d…
Entrywise powers of symmetric matrices preserving positivity, monotonicity or convexity with respect to the Loewner ordering arise in various applications, and have received much attention recently in the literature. Following FitzGerald and Horn [J. Math. Anal. Appl., 1977], it is well-known that there exists a critical exponent beyond which all entrywise powers preserve positive definiteness. Similar phase transition phenomena have also recently been shown by Hiai (2009) to occur for monotonicity and convexity. In this paper, we complete the characterization of all the entrywise powers below…
Given a unitarily invariant ergodic measure on $\infty\times \infty$ Hermitian matrices, it is known that the characteristic function determines (and is determined by) a Polya frequency function $p(t)$. In turn the (finite) measure $dρ(u):=\frac{1}{p(-iu^2)}du$ has the property that the Fourier transform $Z_b$ of $exp(-bu^2)dρ(u)$ is an entire function and has real zeroes, for all $b\ge 0$; this is very close (but not identical) to a classification of such measures due to Newman. This raises the question of whether there is a direct connection between (e.g. the spectrum of) random Hermitian ma…
Entrywise powers of matrices have been well-studied in the literature, and have recently received renewed attention in the regularization of high-dimensional correlation matrices. In this paper, we study powers of positive semidefinite block matrices $(H_{st})_{s,t=1}^n$ with complex entries. We first characterize the powers $α\in\mathbb{R}$ such that the blockwise power map $(H_{st}) \mapsto (H_{st}^α)$ preserves Loewner positivity. The characterization is obtained by exploiting connections with the theory of matrix monotone functions developed by Loewner. Second, we revisit previous work by …
An explicit counterexample shows that contrary to the situation in the special Colombeau algebra, positivity and invertibility cannot be characterized pointwise in algebras of tempered generalized functions. Further a point value characterization of the latter is refined.
Entrywise functions preserving Loewner positivity have been studied by many authors, most notably Schoenberg and Rudin. Following their work, it is known that functions preserving positivity when applied entrywise to positive semidefinite matrices of all dimensions are necessarily analytic with nonnegative Taylor coefficients. When the dimension is fixed, it has been shown by Vasudeva and Horn that such functions are automatically continuous and sufficiently differentiable. A natural refinement of the aforementioned problem consists of characterizing functions preserving positivity under rank …
A new proof is presented of a theorem of L.~Gurvits, which states that the cone of positive block-Toeplitz matrices with matrix entries has no entangled elements. The proof of the Gurvits separation theorem is achieved by making use of the structure of the operator system dual of the operator system $C(S^1)^{(n)}$ of $n\times n$ Toeplitz matrices over the complex field, and by determining precisely the structure of the generators of the extremal rays of the positive cones of the operator systems $C(S^1)^{(n)}\omin\B(\H)$ and $C(S^1)_{(n)}\omin\B(\H)$, where $\H$ is an arbitrary Hilbert space a…
This paper is devoted to the generalization of the theory of total positivity. We say that a linear operator A in R^n is generalized totally positive (GTP), if its jth exterior power preserves a proper cone K_j in the corresponding space for every j = 1, ..., n. We also define generalized strictly totally positive (GSTP) operators. We prove that the spectrum of a GSTP operator is positive and simple, moreover, its eigenvectors are localized in special sets. The existence of invariant cones of finite ranks is shown under some additional conditions. Some new insights and alternative proofs of th…
We show that a formal power series in $2N$ non-commuting indeterminates is a positive non-commutative kernel if and only if the kernel on $N$-tuples of matrices of any size obtained from this series by matrix substitution is positive. We present two versions of this result related to different classes of matrix substitutions. In the general case we consider substitutions of jointly nilpotent $N$-tuples of matrices, and thus the question of convergence does not arise. In the ``convergent'' case we consider substitutions of $N$-tuples of matrices from a neighborhood of zero where the series conv…
Some real functions f induce mean of positive numbers and the matrix monotonicity gives a possibility for means of positive definite matrices. Moreover, such a function f can define linear mapping beta on matrices (which is basic in the constructions of monotone metrics). The present subject is to check the complete positivity of beta in the case of a few concrete functions f. This problem has been motivated by applications in quantum information.
This survey contains a selection of topics unified by the concept of positive semi-definiteness (of matrices or kernels), reflecting natural constraints imposed on discrete data (graphs or networks) or continuous objects (probability or mass distributions). We put emphasis on entrywise operations which preserve positivity, in a variety of guises. Techniques from harmonic analysis, function theory, operator theory, statistics, combinatorics, and group representations are invoked. Some partially forgotten classical roots in metric geometry and distance transforms are presented with comments and …
