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Priyanka Grover

Publications and source records attributed to Priyanka Grover.

15 recordsLinked to original sources
arXiv2026

Properties of best approximations with respect to the Ky Fan $p$-$k$ norm, and the strict spectral approximant of a matrix

Some questions raised in [K. Zi\k{e}tak, {\it From the strict Chebyshev approximant of a vector to the strict spectral approximant of a matrix}, Warsaw : Banach Center Publ., 112 Polish Acad. Sci. Inst. Math. (2017)] are discussed. To do so, the subdifferential set of the Ky Fan $p$-$k$ norm is computed. A characterization for the best approximations with respect to the Ky Fan $p$-$k$ norms is given. Further, necessary and sufficient conditions for $\varepsilon$-Birkhoff orthogonality with respect to the Ky Fan $p$-$k$ norm are also derived.

math.FA↗
arXiv2025

Subdifferential of the $\mathcal{B(H,K)}$ norm, and approximate orthogonality

We present an expression for the right hand derivative of the $\mathcal{B(H,K)}$ norm generalizing the result for $\mathcal{K}=\mathcal{H}$ in [D. J. Ke$\check{\mathrm{c}}$ki$\grave{\mathrm{c}}$, Gateaux derivative of $B(H)$ norm, Proc. Amer. Math. Soc. 133 (2005): 2061--2067]. Using this, we obtain the subdifferential of the $\mathcal{B(H, K)}$ norm. For tuples of operators $\mathbf{A},\mathbf{X}\in$ $\mathcal{B(H, H}^d)$, we give a characterization for $\boldsymbol 0$ to be a best approximation to the subspace $\mathbb C^d \mathbf{X}$, generalizing a similar result for $\mathbb C^d \mathbf{I}$ in [P. Grover, S. Singla, A distance formula for tuples of operators, Linear Algebra Appl. 650 (2022): 267--285]. We define the concept of $\epsilon$-Birkhoff orthogonality to a subspace in a general normed space and derive a characterization in terms of the subdifferential set. Using this, we deduce interesting results for $A\in \mathcal{B(H,K)}$ to be $\epsilon$-Birkhoff orthogonal to a subspace of $\mathcal{B(H,K)}$, when $A$ is compact.

math.FA↗
arXiv2024

Inertia and other properties of the matrix $\left[\beta(i,j)\right]$

Let $\pi(A)$, $\xi(A)$ and $\nu(A)$, respectively, denote the number of positive, zero and negative eigenvalues of the matrix $A$. Then the triplet $(\pi(A), \xi(A), \nu(A))$ is called the \emph{inertia} of $A$ and is denoted by $\textup{Inertia(A)}$. Let $\beta$ be the beta function. The inertia of the matrix $\left[\beta(i,j )\right]$ is shown to be $\left(\frac{n}{2},0,\frac{n}{2}\right)$ if $n$ is even, and $\left(\frac{n+1}{2},0,\frac{n-1}{2}\right)$ if $n$ is odd. %Its connections with Birkhoff-James orthogonality are given. It is also shown that $\left[\beta(i,j)\right]$ is Birkhoff-James orthogonal to the $n\times n$ identity matrix $I$ in the trace norm if and only if $n$ is even. %We prove that the inverse of $\left[{\beta(i,j)}\right]$ is an integer matrix. For $0<\la_1<\cdots<\la_n, 0<\mu_1<\cdots<\mu_n$, it is shown that the matrix $\left[(\beta(\la_i,\mu_j))^m\right]$ is non singular if $\mu_{i+1}-\mu_{i}\in \N$ for all $1\leq i \leq n-1$. It is also shown that if $\mu_{i+1}-\mu_i \in \N$ for $1\leq i\leq n-1$, then for $m\in \mathbb N$, the matrix $\left[\frac{1}{\beta(\la_i,\mu_j)^m}\right]$ is totally positive.

math.CO↗
arXiv2022

A distance formula for tuples of operators

For a tuple of operators $\boldsymbol{A}= (A_1, \ldots, A_d)$, $\text{dist}(\boldsymbol{A}, \mathbb C^d \boldsymbol{I})$ is defined as $\min\limits_{\boldsymbol{z} \in \mathbb C^d} \|\boldsymbol{A-zI}\|$ and $\text{var}_x (\boldsymbol{A})$ as $\|\boldsymbol{A} x\|^2-\sum_{j=1}^d {\big|}\langle x| A_j x\rangle{\big|}^2.$ For a tuple $\boldsymbol{A}$ of commuting normal operators, it is known that $$\text{dist}(\boldsymbol{A}, \mathbb C^d \boldsymbol{I})^2=\sup_{\|x\|=1}\text{var}_x (\boldsymbol{A}).$$ We give an expression for the maximal joint numerical range of a tuple of doubly commuting matrices. Consequently, we obtain that the above distance formula holds for tuples of doubly commuting matrices. We also discuss some general conditions on the tuples of operators for this formula to hold. As a result, we obtain that it holds for tuples of Toeplitz operators as well.

math.FA↗
arXiv2022

Bidiagonal decompositions and total positivity of some special matrices

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.

math.RA↗
arXiv2022

Subdifferential of the joint numerical radius

An expression for the subdifferential of the joint numerical radius is obtained. Its applications to the best approximation problems in the joint numerical radius are discussed.

math.FA↗
arXiv2021

Best approximations, distance formulas and orthogonality in C*-algebras

For a unital $C^*$-algebra $\mathcal A$ and a subspace $\mathcal B$ of $\mathcal A$, a characterization for a best approximation to an element of $\mathcal A$ in $\mathcal B$ is obtained. As an application, a formula for the distance of an element of $\mathcal A$ from $\mathcal B$ has been obtained, when a best approximation of that element to $\mathcal B$ exists. Further, a characterization for Birkhoff-James orthogonality of an element of a Hilbert $C^*$-module to a subspace is obtained.

math.OA↗
arXiv2020

Birkhoff-James orthogonality and applications : A survey

In the last few decades, the concept of Birkhoff-James orthogonality has been used in several applications. In this survey article, the results known on the necessary and sufficient conditions for Birkhoff-James orthogonality in certain Banach spaces are mentioned. Their applications in studying the geometry of normed spaces are given. The connections between this concept of orthogonality, and the Gateaux derivative and the subdifferential set of the norm function are provided. Several interesting distance formulas can be obtained using the characterizations of Birkhoff-James orthogonality, which are also mentioned. In the end, some new results are obtained.

math.FA↗
arXiv2020

Positivity properties of some special matrices

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.

math.FA↗
arXiv2017

Orthogonality to matrix subspaces, and a distance formula

We obtain a necessary and sufficient condition for a matrix $A$ to be Birkhoff-James orthogonal to any subspace $\mathscr W$ of $\mathbb M_n(\mathbb C)$. Using this we obtain an expression for the distance of $A$ from any unital $C^*$ subalgebra of $\mathbb M_n(\mathbb C)$.

math.FA↗
arXiv2017

Derivatives of Multilinear Functions of Matrices

Perturbation or error bounds of functions have been of great interest for a long time. If the functions are differentiable, then the mean value theorem and Taylor's theorem come handy for this purpose. While the former is useful in estimating $\|f(A+X)-f(A)\|$ in terms of $\|X\|$ and requires the norms of the first derivative of the function, the latter is useful in computing higher order perturbation bounds and needs norms of the higher order derivatives of the function. In the study of matrices, determinant is an important function. Other scalar valued functions like eigenvalues and coefficients of characteristic polynomial are also well studied. Another interesting function of this category is the permanent, which is an analogue of the determinant in matrix theory. More generally, there are operator valued functions like tensor powers, antisymmetric tensor powers and symmetric tensor powers which have gained importance in the past. In this article, we give a survey of the recent work on the higher order derivatives of these functions and their norms. Using Taylor's theorem, higher order perturbation bounds are obtained. Some of these results are very recent and their detailed proofs will appear elsewhere.

math.FA↗
arXiv2016

Orthogonality of matrices in the Ky Fan $k$-norms

We obtain necessary and sufficient conditions for a matrix $A$ to be Birkhoff-James orthogonal to another matrix $B$ in the Ky Fan $k$-norms. A characterization for $A$ to be Birkhoff-James orthogonal to any subspace $\mathscr W$ of $\mathbb M(n)$ is also obtained.

math.FA↗
arXiv2015

Derivatives of tensor powers and their norms

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.

math.FA↗