arXiv subjects
Publications and source records attributed to Tanvi Jain.
Let $r$ be any real number and for any $n$ let $p_1,\ldots,p_n$ be distinct positive numbers. A Kwong matrix is the $n\times n$ matrix whose $(i,j)$ entry is $(p_i^r+p_j^r)/(p_i+p_j).$ We determine the signatures of eigenvalues of all such matrices. The corresponding problem for the family of Loewner matrices $\begin{bmatrix}(p_i^r-p_j^r)/(p_i-p_j)\end{bmatrix}$ has been solved earlier.
This article has two interpenetrating motifs. One is an exposition of some major ideas and techniques behind the use of block matrices, and especially their positivity properties. This is done by focussing on one major problem: characterisation of operators whose numerical radius is bounded by one. So, the article could serve as an introduction to that topic as well.
Authentication and authorization of a user's identity are generally done by the service providers or identity providers. However, these centralized systems limit the user's control of their own identity and are prone to massive data leaks due to their centralized nature. We propose a blockchain-based identity management system to authenticate and authorize users using attribute-based access control policies and privacy-preserving algorithms and finally returning the control of a user's identity to the user. Our proposed system would use a private blockchain, which would store the re-certification events and data access and authorization requests for users' identities in a secure, verifiable manner, thus ensuring the integrity of the data. This paper suggests a mechanism to digitize documents such as passports, driving licenses, electricity bills, etc., issued by any government authority or other authority in an immutable and secure manner. The data owners are responsible for authenticating and propagating the users' identities as and when needed using the OpenID Connect protocol to enable single sign-on. We use advanced cryptographic algorithms to provide pseudonyms to the users, thus ensuring their privacy. These algorithms also ensure the auditability of transactions as and when required. Our proposed system helps in mitigating some of the issues in the recent privacy debates. The project finds its applications in citizen transfers, inter-country service providence, banks, ownership transfer, etc. The generic framework can also be extended to a consortium of banks, hospitals, etc.
For every $2n\times 2n$ real positive definite matrix $A,$ there exists a real symplectic matrix $M$ such that $M^TAM=\diag(D,D),$ where $D$ is the $n\times n$ positive diagonal matrix with diagonal entries $d_1(A)\le \cdots\le d_n(A).$ The numbers $d_1(A),\ldots,d_n(A)$ are called the symplectic eigenvalues of $A.$ We derive analogues of Wielandt's extremal principle and multiplicative Lidskii's inequalities for symplectic eigenvalues.
We study the properties of stationary G-chains in terms of their generating functions. In particular, we prove an analogue of the Szegő limit theorem for symplectic eigenvalues, derive an expression for the entropy rate of stationary quantum Gaussian processes, and study the distribution of symplectic eigenvalues of truncated block Toeplitz matrices. We also introduce a concept of symplectic numerical range, analogous to that of numerical range, and study some of its basic properties, mainly in the context of block Toeplitz operators.
Associated with every $2n\times 2n$ real positive definite matrix $A,$ there exist $n$ positive numbers called the symplectic eigenvalues of $A,$ and a basis of $\mathbb{R}^{2n}$ called the symplectic eigenbasis of $A$ corresponding to these numbers. In this paper, we discuss the differentiability (analyticity) of the symplectic eigenvalues and corresponding symplectic eigenbasis for differentiable (analytic) map $t\mapsto A(t),$ and compute their derivatives. We then derive an analogue of Lidskii's theorem for symplectic eigenvalues as an application.
Let $x$ and $y$ be positive $n$-vectors. We show that there exists a $2n\times 2n$ positive definite real matrix whose symplectic spectrum is $y,$ and the symplectic spectrum of whose diagonal is $x$ if and only if $x$ is weakly supermajorised by $y.$
We study ranks of the $r\textrm{th}$ Hadamard powers of doubly nonnegative matrices and show that the matrix $A^{\circ r}$ is positive definite for every $n\times n$ doubly nonnegative matrix $A$ and for every $r>n-2$ if and only if no column of $A$ is a scalar multiple of any other column of $A.$ A particular emphasis is given to the study of rank, positivity and monotonicity of Hadamard powers of rank two, positive semidefinite matrices that have all entries positive.
On the space of positive definite matrices we consider distance functions of the form $d(A,B)=\left[\tr\mathcal{A}(A,B)-\tr\mathcal{G}(A,B)\right]^{1/2},$ where $\mathcal{A}(A,B)$ is the arithmetic mean and $\mathcal{G}(A,B)$ is one of the different versions of the geometric mean. When $\mathcal{G}(A,B)=A^{1/2}B^{1/2}$ this distance is $\|A^{1/2}-B^{1/2}\|_2,$ and when $\mathcal{G}(A,B)=(A^{1/2}BA^{1/2})^{1/2}$ it is the Bures-Wasserstein metric. We study two other cases: $\mathcal{G}(A,B)=A^{1/2}(A^{-1/2}BA^{-1/2})^{1/2}A^{1/2},$ the Pusz-Woronowicz geometric mean, and $\mathcal{G}(A,B)=\exp\big(\frac{\log A+\log B}{2}\big),$ the log Euclidean mean. With these choices $d(A,B)$ is no longer a metric, but it turns out that $d^2(A,B)$ is a divergence. We establish some (strict) convexity properties of these divergences. We obtain characterisations of barycentres of $m$ positive definite matrices with respect to these distance measures.
We present several theorems on strict and strong convexity, and higher order differential formulae for sandwiched quasi-relative entropy (a parametrised version of the classical fidelity). These are crucial for establishing global linear convergence of the gradient projection algorithm for optimisation problems for these functions. The case of the classical fidelity is of special interest for the multimarginal optimal transport problem (the $n$-coupling problem) for Gaussian measures.
Positivity properties of the Hadamard powers of the matrix $\begin{bmatrix}1+x_ix_j\end{bmatrix}$ for distinct positive real numbers $x_1,\ldots,x_n$ and the matrix $\begin{bmatrix}|\cos((i-j)π/n)|\end{bmatrix}$ are studied. In particular, it is shown that $\begin{bmatrix}(1+x_ix_j)^r\end{bmatrix}$ is not positive semidefinite for any positive real number $r<n-2$ that is not an integer, and $\begin{bmatrix}|\cos((i-j)π/n)|^r\end{bmatrix}$ is positive semidefinite for every odd integer $n\ge 3$ and $n-3\le r<n-2.$
If $A$ is a $2n \times 2n$ real positive definite matrix, then there exists a symplectic matrix $M$ such that $M^TAM = \left [ \begin{array}{cc} D & O \\ O & D \end{array} \right ]$ where $D= \diag (d_1 (A), \ldots, d_n(A))$ is a diagonal matrix with positive diagonal entries, which are called the symplectic eigenvalues of $A.$ In this paper we derive several fundamental inequalities about these numbers. Among them are relations between the symplectic eigenvalues of $A$ and those of $A^t,$ between the symplectic eigenvalues of $m$ matrices $A_1, \ldots, A_m$ and of their Riemannian mean, a perturbation theorem, some variational principles, and some inequalities between the symplectic and ordinary eigenvalues.
Let $p_1<p_2<\cdots<p_n$ be positive real numbers. It is shown that the matrix whose $i,j$ entry is $(p_i+p_j)^{p_i+p_j}$ is infinitely divisible, nonsingular and totally positive.
We prove majorization inequalities for different means of positive definite matrices. These include the Cartan mean (the Karcher mean), the log Euclidean mean, the Wasserstein mean and the power mean.
The metric $d(A,B)=\left[ \tr\, A+\tr\, B-2\tr(A^{1/2}BA^{1/2})^{1/2}\right]^{1/2}$ on the manifold of $n\times n$ positive definite matrices arises in various optimisation problems, in quantum information and in the theory of optimal transport. It is also related to Riemannian geometry. In the first part of this paper we study this metric from the perspective of matrix analysis, simplifying and unifying various proofs. Then we develop a theory of a mean of two, and a barycentre of several, positive definite matrices with respect to this metric. We explain some recent work on a fixed point iteration for computing this Wasserstein barycentre. Our emphasis is on ideas natural to matrix analysis.
The norm of the $m$th derivative of the map that takes an operator to its $k$th antisymmetric tensor power is evaluated. The case $m=1$ has been studied earlier by Bhatia and Friedland [R. Bhatia and S. Friedland, Variation of Grassman powers and spectra, Linear Algebra and its Applications, 40:1--18, 1981]. For this purpose a multilinear version of a theorem of Russo and Dye is proved: it is shown that a positive $m$-linear map between $C^{\ast}$-algebras attains its norm at the $m$-tuple $(I, \, I, ..., I).$ Expressions for derivatives of the maps that take an operator to its $k$th tensor power and $k$th symmetric tensor power are also obtained. The norms of these derivatives are computed. Derivatives of the map taking a matrix to its permanent are also evaluated.
Given positive numbers p_1 < p_2 < ... < p_n, and a real number r let L_r be the n by n matrix with its (i,j) entry equal to (p_i^r-p_j^r)/(p_i-p_j). A well-known theorem of C. Loewner says that L_r is positive definite when 0 < r < 1. In contrast, R. Bhatia and J. Holbrook, (Indiana Univ. Math. J, 49 (2000) 1153-1173) showed that when 1 < r < 2, the matrix L_r has only one positive eigenvalue, and made a conjecture about the signatures of eigenvalues of L_r for other r. That conjecture is proved in this paper.
We study the function $(1 - \|x\|)\slash (1 - \|x\|^r),$ and its reciprocal, on the Euclidean space $\mathbb{R}^n,$ with respect to properties like being positive definite, conditionally positive definite, and infinitely divisible.