Ashoka University is a private university located in Sonipat, Haryana, India, focusing on liberal arts and sciences. It was established in 2014.&nbsp;It is guided by an academic council of academics and scholars from around the world. The council sets the university&#39;s academic standard, advising on matters of curriculum, faculty hiring and research.&nbsp;

Ashoka University offers a full-time 3 (+ 1) year undergraduate programme. The University comprises of&nbsp;2500 plus students, with a mix of both national and international students.&nbsp; &nbsp;

&nbsp;

&nbsp;

Ashoka University

This article has two interpenetrating motifs. One is an exposition of some major ideas and techniques behind the use of <a href="https://www.sciencedirect.com/topics/mathematics/block-matrix">block matrices</a>, and especially their positivity <a href="https://www.sciencedirect.com/topics/mathematics/sigma-property">properties</a>. 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.

Department of Mathematics

World University root node

Linear Algebra and Its Applications

The numerical radius and positivity of block matrices

Let&nbsp;&lambda;j&nbsp;and&nbsp;ajj,&nbsp;1&le;j&le;n, be the eigenvalues and the diagonal entries of a <a href="https://www.sciencedirect.com/topics/mathematics/hermitian-matrix">Hermitian matrix</a> A, both enumerated in the increasing order. We prove some inequalities that are stronger than the Schur majorisation inequalities&nbsp;&sum;j=1r&lambda;j&le;&sum;j=1rajj,&nbsp;1&le;r&le;n.

Fulltext

Extensions of the Schur majorisation inequalities

Let f be a transformation on the Euclidean space such that f(0) = 0, and f preserves distances. Then f is linear, and this makes it easier to analyze f. The Mazur—Ulam theorem generalizes this to maps between real normed linear spaces. We discuss this theorem and its proofs.

Resonance

Isometries and Linearity

Indian Journal of Pure and Applied Mathematics

Eigenvalues and Diagonal Elements

We study the properties of stationary G-chains in terms of their generating functions. In particular, we prove an analogue of the Szego 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. © 2021 European Mathematical Society Publishing House. All rights reserved.

Journal of Spectral Theory

A Szego type theorem and distribution of symplectic eigenvalues

If A is a real 2n×2n positive definite matrix, then there exists a symplectic matrix M such that MTAM=diag(D,D), where D is a positive diagonal matrix with diagonal entries d1(A)⩽⋯⩽dn(A). We prove a maxmin principle for dk(A) akin to the classical Courant–Fisher–Weyl principle for Hermitian eigenvalues and use it to derive an analogue of the Weyl inequality di+j−1(A+B)⩾di(A)+dj(B).

Canadian Mathematical Bulletin

Variational principles for symplectic eigenvalues

Let x and y be positive n-vectors. We show that there exists a 2n×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. © 2020 Elsevier Inc.

A Schur-Horn theorem for symplectic eigenvalues

On the space of positive definite matrices, we consider distance functions of the form 𝑑(𝐴,𝐵)=[tr𝒜(𝐴,𝐵)−tr𝒢(𝐴,𝐵)]1/2,<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mo>[</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">t</mi><mi mathvariant="normal">r</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="script">A</mi></mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo><mo>−</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">t</mi><mi mathvariant="normal">r</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="script">G</mi></mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo><mo>]</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn></mrow></msup><mo>,</mo></math>&nbsp;where 𝒜(𝐴,𝐵)<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="script">A</mi></mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></math>&nbsp;is the arithmetic mean and 𝒢(𝐴,𝐵)<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="script">G</mi></mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></math>&nbsp;is one of the different versions of the geometric mean. When 𝒢(𝐴,𝐵)=𝐴1/2𝐵1/2<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="script">G</mi></mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>A</mi><mrow class="MJX-TeXAtom-ORD"><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn></mrow></msup><msup><mi>B</mi><mrow class="MJX-TeXAtom-ORD"><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn></mrow></msup></math>&nbsp;this distance is ‖𝐴1/2−𝐵1/2‖2,<math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">‖</mo><msup><mi>A</mi><mrow class="MJX-TeXAtom-ORD"><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn></mrow></msup><mo>−</mo><msup><mi>B</mi><mrow class="MJX-TeXAtom-ORD"><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn></mrow></msup><msub><mo fence="false" stretchy="false">‖</mo><mn>2</mn></msub><mo>,</mo></math>&nbsp;and when 𝒢(𝐴,𝐵)=(𝐴1/2𝐵𝐴1/2)1/2<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="script">G</mi></mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msup><mi>A</mi><mrow class="MJX-TeXAtom-ORD"><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn></mrow></msup><mi>B</mi><msup><mi>A</mi><mrow class="MJX-TeXAtom-ORD"><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn></mrow></msup><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn></mrow></msup></math>&nbsp;it is the Bures–Wasserstein metric. We study two other cases: 𝒢(𝐴,𝐵)=𝐴1/2(𝐴−1/2𝐵𝐴−1/2)1/2𝐴1/2,<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="script">G</mi></mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>A</mi><mrow class="MJX-TeXAtom-ORD"><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn></mrow></msup><mo stretchy="false">(</mo><msup><mi>A</mi><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn></mrow></msup><mi>B</mi><msup><mi>A</mi><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn></mrow></msup><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn></mrow></msup><msup><mi>A</mi><mrow class="MJX-TeXAtom-ORD"><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn></mrow></msup><mo>,</mo></math>&nbsp;the Pusz–Woronowicz geometric mean, and 𝒢(𝐴,𝐵)=exp(log𝐴+log𝐵2),<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="script">G</mi></mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mi>exp</mi><mo>⁡</mo><mstyle scriptlevel="0"><mrow class="MJX-TeXAtom-ORD"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow></mstyle><mfrac><mrow><mi>log</mi><mo>⁡</mo><mi>A</mi><mo>+</mo><mi>log</mi><mo>⁡</mo><mi>B</mi></mrow><mn>2</mn></mfrac><mstyle scriptlevel="0"><mrow class="MJX-TeXAtom-ORD"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mstyle><mo>,</mo></math>&nbsp;the log Euclidean mean. With these choices, d(A,&nbsp;B) is no longer a metric, but it turns out that 𝑑2(𝐴,𝐵)<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>d</mi><mn>2</mn></msup><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></math>&nbsp;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. One of these leads to a new interpretation of a power mean introduced by Lim and Palfia, as a barycentre. The other uncovers interesting relations between the log Euclidean mean and relative entropy.

Letters in Mathematical Physics

Matrix versions of the Hellinger distance

On the space of positive definite matrices, we consider distance functions of the form d(A, B) = [tr A(A, B) - tr G(A, B)] 1 / 2, where A(A, B) is the arithmetic mean and G(A, B) is one of the different versions of the geometric mean. When G(A, B) = A1 / 2B1 / 2 this distance is ‖ A1 / 2- B1 / 2‖ 2, and when G(A,B)=(A1/2BA1/2)1/2 it is the Bures–Wasserstein metric. We study two other cases: G(A,B)=A1/2(A-1/2BA-1/2)1/2A1/2, the Pusz–Woronowicz geometric mean, and G(A,B)=exp(logA+logB2), the log Euclidean mean. With these choices, d(A, B) is no longer a metric, but it turns out that d2(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. One of these leads to a new interpretation of a power mean introduced by Lim and Palfia, as a barycentre. The other uncovers interesting relations between the log Euclidean mean and relative entropy. © 2019, Springer Nature B.V.

Matrix versions of the Hellinger distance

We show that for any two n × n matrices X and Y we have the inequality sj2(I + XY) ≤ λ j((I + X∗X)(I + Y∗Y)), where sj(T) and λj(T) denote the decreasingly ordered singular values and eigenvalues of T. As an application, we show that for 2n × 2n real positive definite matrices the symplectic eigenvalues dj, under some special conditions, satisfy the inequality dj(A + B) ≥ dj(A) + d1(B). © 2019 World Scientific Publishing Company.

International Journal of Mathematics

Some inequalities for eigenvalues and symplectic eigenvalues of positive definite matrices

The metric d(A,B)=trA+trB−2tr(A1∕2BA1∕2)1∕2 1∕2 on the manifold of n×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. © 2018 Elsevier GmbH

Expositiones Mathematicae

On the Bures–Wasserstein distance between positive definite matrices

We consider the manifold of positive definite matrices endowed with the Fisher Riemannian metric and some other distances commonly used in information theory. We show that for each of them the best approximant to A from the unitary orbit of another matrix B commutes with A. © 2018 Elsevier Inc.

Procrustes problems in Riemannian manifolds of positive definite matrices

Correction to: Matrix versions of the Hellinger distance (Letters in Mathematical Physics, (2019), 109, 8, (1777-1804), 10.1007/s11005-019-01156-0)

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. © 2018 Elsevier Inc.

Inequalities for the Wasserstein mean of positive definite matrices

We present several theorems on strict and strong convexity, and higher order differential formulae for sandwiched quasi-relative entropy (a parametrized version of the classical fidelity). These are crucial for establishing global linear convergence of the gradient projection algorithm for optimization 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. © 2018 World Scientific Publishing Company.

Reviews in Mathematical Physics

Strong convexity of sandwiched entropies and related optimization problems

The vibrating string problem is the source of much mathematics and physics. This article describes Lagrange’s formulation of a discretised version of the problem and its solution. This is also the first instance of an eigenvalue problem. © 2017, Indian Academy of Sciences.