Eigenvalues of functions of orthogonal projectors

By representing two orthogonal projectors in a finite dimensional vector space as partitioned matrices, several characterizations concerning eigenvalues of various functions of the pair are obtained. These results substantially extend the ones already available in the literature. Additionally, some related results dealing with the functions of a pair of orthogonal projectors are provided, with the emphasis laying on the problem of invertibility.     

Oblique projectors and group involutory matrices

In this work, conditions characterizing when a lineal combination of two projectors of the same order is a group involutory matrix are analyzed. In particular, a representation of an involutory matrix as a sum or difference of two projectors is given. In addition, some results are applied to control problems. An application to obtain a state feedback control is given.     

Revisitation of the product of two orthogonal projectors

Several results involving a product of two orthogonal projectors (i.e., Hermitian idempotent matrices) are established by exploring a representation of the product as a partitioned matrix. These results concern, for instance, rank, trace, range, null space, generalized inverses, and spectral properties of the product and its various functions. Particular attention is paid to the conditions equivalent to the requirement that the product of two orthogonal projectors is an orthogonal projector itself, and these characterizations refer to such known classes of matrices as Hermitian, involutory, normal, star-dagger, unitary as well as partial isometries and semi-orthogonal projectors. Moreover, some results dealing with the notions of parallel sum and spectral norm are obtained. The variety of problems considered shows that the approach utilized in the paper provides a powerful tool of wide applicability.     

On oblique projection, rank additivity and the moore-penrose inverse of the sum of two matrices

The existence of certain oblique projectors is revealed by deriving onto-and along-space of a specific projector, namely the Moore-Penrose inverse of the product of two orthogonal projectors. The results are used to create a formula for the Moore-Penrose in-verse of the sum of two rank additive matrices, and to demonstrate relationships with already known facts.     

Representation of projectors involving Minkowski inverse in Minkowski space

A certain class of results about the different representations of Oblique projectors is present in the literature. These results represent Oblique projectors as the functions of orthogonal projectors with given onto and along spaces. But these results are valid under the restriction that the functions of orthogonal projectors involved are invertible. In this paper we extend and generalize these results. The extension lies in making a transition from Euclidean space to Minkowski space M and the generalization is obtain by voiding the invertibility condition and use of the Minkowski inverse. Furthermore, the nobility lies in utilizing the m-projectors instead of the regular orthogonal projectors.     

行分块矩阵值域上正交投影的展开式

We give in this note some expansion formulas for the orthogonal projectors onto the range of the row block matrix [ A, B ], and use the expansion formulas to examine relations among the orthogonal projectors onto the ranges of A, B and [A, B]. In particular, we present some identifying conditions for a pair of orthogonal projectors of the same size to commute.     

On the Decompositions of a Quantum State

We classify all the decompositions of a quantum state as a weighted sum of one dimensional projectors. In particular we describe explicitly the set of irreducible decompositions. The physical interest in this problem rests on the possibility of interpreting the decomposition in terms of a classical mixture.     

Column space equalities for orthogonal projectors

In their paper [Y. Tian, G.P.H. Styan, Rank equalities for idempotent and involutory matrices. Linear Algebra Appl. 335 (2001) 101-117], Tian and Styan established several rank equalities involving a pair of idempotent matrices P and Q. Subsequently, these results are reinvestigated from the point of view of the following question: provided that idempotent P, Q are Hermitian, which relationships given in the aforementioned paper remain valid when ranks are replaced with column spaces? Simultaneously, some related results are established, which shed additional light on the links between subspaces attributed to various functions of a pair of orthogonal projectors.     

On the Limits of Lagrange Projectors

This article addresses a question of Carl de Boor (In: Constructive theory of functions, Varna 2005, pp. 51–63, Marin Drinov Academic, Sofia, [2006]): What ideal projectors are the limits of Lagrange projectors? The results of this paper answer the question in the sense that for every ideal projector P, we prescribe finitely many computations that determine whether the projector P is a limit of Lagrange projectors.     

Optimization over Young diagrams

Journal of Algebraic Combinatorics - We consider the problem of finding a Young diagram minimizing the sum of evaluations of a given pair of functions on the parts of the associated pair of...     

Subgradient Projectors: Extensions,Theory, and Characterizations

Subgradient projectors play an important role in optimization and for solving convex feasibility problems. For every locally Lipschitz function, we can define a subgradient projector via generalized subgradients even if the function is not convex. The paper consists of three parts. In the first part, we study basic properties of subgradient projectors and give characterizations when a subgradient projector is a cutter, a local cutter, or a quasi-nonexpansive mapping. We present global and local convergence analyses of subgradent projectors. Many examples are provided to illustrate the theory. In the second part, we investigate the relationship between the subgradient projector of a prox-regular function and the subgradient projector of its Moreau envelope. We also characterize when a mapping is the subgradient projector of a convex function. In the third part, we focus on linearity properties of subgradient projectors. We show that, under appropriate conditions, a linear operator is a subgradient projector of a convex function if and only if it is a convex combination of the identity operator and a projection operator onto a subspace. In general, neither a convex combination nor a composition of subgradient projectors of convex functions is a subgradient projector of a convex function.     

On the existence of certain error formulas for a special class of ideal projectors

In this paper, we focus on a special class of ideal projectors. With the aid of algebraic geometry, we prove that for this special class of ideal projectors, there exist “good” error formulas as defined by C. de Boor. Furthermore, we completely analyze the properties of the interpolation conditions matched by this special class of ideal projectors, and show that the ranges of this special class of ideal projectors are the minimal degree interpolation spaces with regard to their associated interpolation conditions.     

群体多指标决策的偏好集结方法

研究多个指标条件下,利用个体决策结果形成群体一致偏好的方法、假设个体有加性效用函数,将个体多指标效用函数表示成单个指标评价函数的加权和,群体指标评价函数表示成个体指标评价函数的加权和.通过协商指标权重、指标评价函数、支付意愿三个参数,成对个体达成双方一致.提出了(n-1)对个体之间达成双方一致,从而得出群体效用函数的决策方法,这种分析框架同样可以扩展到联盟协商一致中.     

A General Global Optimization Approach for Solving Location Problems in the Plane

We propose a general approach for constructing bounds required for the “Big Triangle Small Triangle” (BTST) method for the solution of planar location problems. Optimization problems, which constitute a sum of individual functions, each a function of the Euclidean distance to a demand point, are analyzed and solved. These bounds are based on expressing each of the individual functions in the sum as a difference between two convex functions of the distance, which is not the same as convex functions of the location. Computational experiments with nine different location problems demonstrated the effectiveness of the proposed procedure.     

Double <Emphasis Type="Italic">SO</Emphasis>(2, 1)-integrals and formulas for Whittaker functions

With the help of some double integral bilinear functionals with homogeneous kernels defined on a pair of representation spaces of the group SO(2, 1) we obtain some functional relations for Whittaker functions and calculate the sum of one series of Gauss hypergeometric functions converging to a Whittaker function.     

Comultiplications with Meson Algebras

The mesonic interior multipliciations have suggested the existence of mesonic comultiplications. The proof of their existence has motivated an intensive study of the meson algebra of an orthogonal sum, revealing its twisted gradations and its canonical projectors. The traditional applications of comultiplications follow. A final application to mesonic deformations corroborates the close relationship between Clifford algebras and meson algebras.     

Master equations of Pointer hamiltonians and decoherence

In this article, we present an elementary derivation of the notion of completely positive dynamics which is extremely general. The hypothesis is that the correlations of the spectral projectors of the bath hamiltonian are δ functions. We derive a decoherence result for pointer hamiltonian in the basis of the pointer observable.     

Product of random projections,Jacobi ensembles and universality problems arising from free probability

We consider the product of two independent randomly rotated projectors. The square of its radial part turns out to be distributed as a Jacobi ensemble. We study its global and local properties in the large dimension scaling relevant to free probability theory. We establish asymptotics for one point and two point correlation functions, as well as properties of largest and smallest eigenvalues. B.C. is currently a JSPS postdoctoral fellow     

Necessary Covariance Conditions for a One-Field Lax Pair

We study the covariance with respect to Darboux transformations of polynomial differential and difference operators with coefficients given by functions of one basic field. In the scalar (Abelian) case, the functional dependence is established by equating the Frechet differential (the first term of the Taylor series on the prolonged space) to the Darboux transform; a Lax pair for the Boussinesq equation is considered. For a pair of generalized Zakharov-Shabat problems (with differential and shift operators) with operator coefficients, we construct a set of integrable nonlinear equations together with explicit dressing formulas. Non-Abelian special functions are fixed as the fields of the covariant pairs. We introduce a difference Lax pair, a combined gauge-Darboux transformation, and solutions of the Nahm equations.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 1, pp. 122–132, July, 2005.     

投影算子组的联合谱

本文研究Hilbert空间H上投影算子组的联合谱.首先通过计算给出正则投影对的联合谱,进而给出一般的投影算子对的联合谱.本文还对两个投影算子的和与差的可逆性给出一些等价刻画.特别地,当P和Q为正则投影对时,本文通过计算算子组[I, P, Q]的联合谱来给出σ(P+Q)和σ(P-Q)的具体刻画.反过来,本文证明两类具有特定形式的复数集分别是Hilbert空间上正则投影对的和与差的谱.本文也给出一般的投影算子对的和与差的谱.最后,本文计算特定条件下的3个投影算子组[P, Q, R]的联合谱.