Matrix Convexity Research Articles

Correction to: Ruminations on matrix convexity and the strong subadditivity of quantum entropy

Correction to: Ruminations on matrix convexity and the strong subadditivity of quantum entropy

Matrix perspectives and matrix convexity

In this article, following [4,3], we consider matrix perspectives and show that the n-convexity of a function f on an interval I⊂(0,∞) is equivalent to that of the function g(t)=tf(t−1) on I−1.

Ruminations on matrix convexity and the strong subadditivity of quantum entropy

The familiar second derivative test for convexity, combined with resolvent calculus, is shown to yield a useful tool for the study of convex matrix-valued functions. We demonstrate the applicability of this approach on a number of theorems in this field. These include convexity principles which play an essential role in the Lieb–Ruskai proof of the strong subadditivity of quantum entropy.

Dilation theory in finite dimensions and matrix convexity

We establish a finite-dimensional version of the Arveson-Stinespring dilation theorem for unital completely positive maps on operator systems. This result can be seen as a general principle to deduce finite-dimensional dilation theorems from their classical infinite-dimensional counterparts. In addition to providing unified proofs of known finite-dimensional dilation theorems, we establish finite-dimensional versions of Agler's theorem on rational dilation on an annulus, of Berger's dilation theorem for operators of numerical radius at most $1$, and of the Putinar-Sandberg numerical range dilation theorem. As a key tool, we prove versions of Carath\'{e}odory's and of Minkowski's theorem for matrix convex sets.

Finite Dimensionality in the Non-commutative Choquet Boundary: Peaking Phenomena and C*-Liminality

Abstract We explore the finite-dimensional part of the non-commutative Choquet boundary of an operator algebra. In other words, we seek finite-dimensional boundary representations. Such representations may fail to exist even when the underlying operator algebra is finite dimensional. Nevertheless, we exhibit mechanisms that detect when a given finite-dimensional representation lies in the Choquet boundary. Broadly speaking, our approach is topological and requires identifying isolated points in the spectrum of the $\textrm{C}^{\ast }$-envelope. This is accomplished by analyzing peaking representations and peaking projections, both of which being non-commutative versions of the classical notion of a peak point for a function algebra. We also connect this question with the residual finite dimensionality of the $\textrm{C}^{\ast }$-envelope and to a stronger property that we call $\textrm{C}^{\ast }$-liminality. Recent developments in matrix convexity allow us to identify a pivotal intermediate property, whereby every matrix state is locally finite dimensional.

Three-Phase Four-Wire OPF-Based Collaborative Control of PV Inverter and ESS for Low-Voltage Distribution Networks With High Proportion PVs

The use of photovoltaic reactive power and energy storage active power can solve the problems of voltage violation, network loss, and three-phase unbalance caused by photovoltaic connection to low-voltage distribution networks. However, the three-phase four-wire structure of the low-voltage distribution network brings difficulties to power flow calculation. In order to achieve photovoltaic utilization through optimal power flow, a photovoltaic-energy storage collaborative control method for low-voltage distribution networks based on the optimal power flow of a three-phase four-wire system is proposed. Considering the amplitude and phase angle of voltage and current, a three-phase four-wire node admittance matrix was used to establish the network topology of the low-voltage distribution network. Also, to minimize the network loss, the three-phase unbalance and voltage deviation. a multi-objective optimization model based on three-phase four-wire network topology was established considering the voltage constraints, reverse power flow constraints and neutral line current constraints. Through improving the node admittance matrix and model convexity, the complexity of solving the problem is reduced. The CPLEX algorithm package was used to solve the problem. Based on a 21-bus three-phase four-wire low-voltage distribution network, a 24-h multi-period simulation was undertaken to verify the feasibility and effectiveness of the proposed scheme.

On the local theory of prescribed Jacobian equations revisited

<abstract> In this paper we revisit our previous study of the local theory of prescribed Jacobian equations associated with generating functions, which are extensions of cost functions in the theory of optimal transportation. In particular, as foreshadowed in the earlier work, we provide details pertaining to the relaxation of a monotonicity condition in the underlying convexity theory and the consequent classical regularity. Taking advantage of recent work of Kitagawa and Guillen, we also extend our classical regularity theory to the weak form A3w of the critical matrix convexity conditions. </abstract>

Loewner's theorem in several variables

In this paper we establish a multivariable non-commutative generalization of Loewner's theorem characterizing operator monotone functions as real functions admitting analytic continuation mapping the upper complex half-plane into itself. The non-commutative several variable theorem proved here represents several variable operator monotone functions of positive operator tuples, not a priori assumed to be analytic or continuous, as a Schur complement of a positive linear matrix pencil. To achieve this we establish the abstract integral formula for these functions using only matrix convexity and LMIs that represents operator monotone and operator concave free functions as a conditional expectation of a Schur complement of a linear matrix pencil on a tensor product operator algebra. The results can be applied to any of the various multivariable operator means that have been constructed in the last decades or so, including the Karcher mean, to obtain an explicit, closed formula for these operator means.

Local characterizations for the matrix monotonicity and convexity of fixed order

We establish local characterizations of matrix monotonicity and convexity of fixed order by giving integral representations connecting the Loewner and Kraus matrices, previously known to characterize these properties, to respective Hankel matrices. Our results are new already in the general case of matrix convexity, and our approach significantly simplifies the corresponding work on matrix monotonicity. We also obtain an extension of the original characterization for matrix convexity by Kraus and tighten the relationship between monotonicity and convexity.

Matrix convex sets without absolute extreme points

This article shows the existence of a class of closed bounded matrix convex sets which do not have absolute extreme points. The sets we consider are noncommutative sets, KX, formed by taking matrix convex combinations of a single tuple X. In the case that X is a tuple of compact operators with no nontrivial finite dimensional reducing subspaces, KX is a closed bounded matrix convex set with no absolute extreme points.A central goal in the theory of matrix convexity is to find a natural notion of an extreme point in the dimension free setting which is minimal with respect to spanning. Matrix extreme points are the strongest type of extreme point known to span matrix convex sets; however, they are not necessarily the smallest set which does so. Absolute extreme points, a more restricted type of extreme points that are closely related to Arveson's boundary, enjoy a strong notion of minimality should they span. This result shows that matrix convex sets may fail to be spanned by their absolute extreme points.

Extremal matrix states on tensor product of $$C^{*}$$ C ∗ -algebras

A classic result of Namioka and Phelps on the extreme points of the state space on the tensor product of order unit spaces is generalized to the setting of matrix convexity. We show that the matrix extreme points of the matrix state space on the tensor product of two unital $$C^*$$ -algebras, at least one of them is of type $$\mathrm {I}$$ , are exactly the isometric orbit of the restrictions, on the exponentially unitary orbit of the $$C^*$$ -algebraic tensor product, of the tensor products of the matrix extreme points of matrix state spaces of the two $$C^*$$ -algebras.

Loewner matrices of matrix convex and monotone functions

The matrix convexity and the matrix monotony of a real $C^1$ function $f$ on $(0,\infty)$ are characterized in terms of the conditional negative or positive definiteness of the Loewner matrices associated with $f$, $tf(t)$, and $t^2f(t)$. Similar characterizations are also obtained for matrix monotone functions on a finite interval $(a,b)$.

A matrix convexity approach to some celebrated quantum inequalities

Some of the important inequalities associated with quantum entropy are immediate algebraic consequences of the Hansen-Pedersen-Jensen inequalities. A general argument is given using matrix perspectives of operator convex functions. A matrix analogue of Maréchal's extended perspectives provides additional inequalities, including a p + q < or = 1 result of Lieb.

Non-commutative partial matrix convexity

Let p be a polynomial in the non-commuting variables (a, x) = (a1, . . . , ag, x1, . . . , xg). If p is convex in the variables x, then p has degree two in x and moreover, p has the form p = L + Λ Λ, where L has degree at most one in x and Λ is a (column) vector which is linear in x, so that Λ Λ is a both sum of squares and homogeneous of degree two. Of course the converse is true also. Further results involving various convexity hypotheses on the x and a variables separately are presented.

Differential analysis of matrix convex functions

We analyze matrix convex functions of a fixed order defined in a real interval by differential methods as opposed to the characterization in terms of divided differences given by Kraus [F. Kraus, Über konvekse Matrixfunktionen, Math. Z. 41 (1936) 18–42]. We obtain for each order conditions for matrix convexity which are necessary and locally sufficient, and they allow us to prove the existence of gaps between classes of matrix convex functions of successive orders, and to give explicit examples of the type of functions contained in each of these gaps. The given conditions are shown to be also globally sufficient for matrix convexity of order two. We finally introduce a fractional transformation which connects the set of matrix monotone functions of each order n with the set of matrix convex functions of the following order n + 1.

Analysis of Matrix-Convex Functions

The properties of matrix-convex functions, which may take values from an expanded numerical axis, are analyzed. Matrix convexity is specified by a pair of matrices, a final set of matrices, or an infinite family of matrices. The results obtained are used in the theory of differential games.

Convex Noncommutative Polynomials Have Degree Two or Less

A polynomial p (with real coefficients) in noncommutative variables is matrix convex provided \[ p(tX+(1-t)Y) \le tp(X)+(1-t)p(Y) \] for all $0 \le t \le 1$ and for all tuples X=(X1 ,. . .,Xg and Y=(Y1 ,. . .,Yg) of symmetric matrices on a common finite dimensional vector space of a sufficiently large dimension (depending upon p). The main result of this paper is that every matrix convex polynomial has degree two or less. More generally, the polynomial p has degree at most two if convexity holds only for all matrices X and Y in an "open set." An analogous result for nonsymmetric variables is also obtained. Matrix convexity is an important consideration in engineering system theory. This motivated our work, and our results suggest that matrix convexity in conjunction with a type of "system scalability" produces surprisingly heavy constraints.

A stronger version of matrix convexity as applied to functions of Hermitian matrices

A stronger version of matrix convexity as applied to functions of Hermitian matrices

Matrix Convexity: Operator Analogues of the Bipolar and Hahn–Banach Theorems

Several basic results of convexity theory are generalized to the “quantized” matrix convex sets of Wittstock. These include the Bipolar theorem, a gauge version of the Hahn–Banach theorem, and the existence theorem for support functionals.

Some convex and monotone matrix functions

We prove some matrix monotonicity and matrix convexity properties for functions derived from operator connections. As a consequence, we prove that a certain product of matrix monotone functions is a matrix monotone function. Similar results are proved for matrix convex functions. These results include some known results for matrix monotone and matrix convex functions.