On commutativity of projectors

The purpose of this paper is to revisit two problems discussed previously in the literature, both related to the commutativity property P₁P₂ = P₂P₁, where P₁ and P₂ denote projectors (i.e., idempotent matrices). The first problem was considered by Baksalary et al. [J.K. Baksalary, O.M. Baksalary, T. Szulc, A property of orthogonal projectors, Linear Algebra Appl. 354 (2002) 35-39], who have shown that if P₁ and P₂ are orthogonal projectors (i.e., Hermitian idempotent matrices), then in all nontrivial cases a product of any length having P₁ and P₂ as its factors occurring alternately is equal to another such product if and only if P₁ and P₂ commute. In the present paper a generalization of this result is proposed and validity of the equivalence between commutativity property and any equality involving two linear combinations of two any length products having orthogonal projectors P₁ and P₂ as their factors occurring alternately is investigated. The second problem discussed in this paper concerns specific generalized inverses of the sum P₁ + P₂ and the difference P₁ − P₂ of (not necessary orthogonal) commuting projectors P₁ and P₂. The results obtained supplement those provided in Section 4 of Baksalary and Baksalary [J.K. Baksalary, O.M. Baksalary, Commutativity of projectors, Linear Algebra Appl. 341 (2002) 129-142].     

Particular formulae for the Moore-Penrose inverse of a columnwise partitioned matrix

An essential part of Cegielski’s [Obtuse cones and Gram matrices with non-negative inverse, Linear Algebra Appl. 335 (2001) 167-181] considerations of some properties of Gram matrices with nonnegative inverses, which are pointed out to be crucial in constructing obtuse cones, consists in developing some particular formulae for the Moore-Penrose inverse of a columnwise partitioned matrix A = (A₁ : A₂) under the assumption that it is of full column rank. In the present paper, these results are generalized and extended. The generalization consists in weakening the assumption mentioned above to the requirement that the ranges of A₁ and A₂ are disjoint, while the extension consists in introducing the conditions referring to the class of all generalized inverses of A.     

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.     

The nullity and rank of linear combinations of idempotent matrices

Baksalary and Baksalary [J.K. Baksalary, O.M. Baksalary, Nonsingularity of linear combinations of idempotent matrices, Linear Algebra Appl. 388 (2004) 25-29] proved that the nonsingularity of P₁ + P₂, where P₁ and P₂ are idempotent matrices, is equivalent to the nonsingularity of any linear combinations c₁P₁ + c₂P₂, where c₁, c₂ ≠ 0 and c₁ + c₂ ≠ 0. In the present note this result is strengthened by showing that the nullity and rank of c₁P₁ + c₂P₂ are constant. Furthermore, a simple proof of the rank formula of Groß and Trenkler [J. Groß, G. Trenkler, Nonsingularity of the difference of two oblique projectors, SIAM J. Matrix Anal. Appl. 21 (1999) 390-395] is obtained.     

Invariance properties of a triple matrix product involving generalized inverses

Given complex-valued matrices A, B and C of appropriate dimensions, this paper investigates certain invariance properties of the product AXC with respect to the choice of X, where X is a generalized inverse of B. Different types of generalized inverses are taken into account. The purpose of the paper is three-fold: First, to review known results scattered in the literature, second, to demonstrate the connection between invariance properties and the concept of extremal ranks of matrices, and third, to add new results related to the topic.     

Polynomials satisfied by two linked matrices

Polynomials in two variables, evaluated at A and with A being a square complex matrix and being its transform belonging to the set {A⁼, A^†, A^∗}, in which A⁼, A^†, and A^∗ denote, respectively, any reflexive generalized inverse, the Moore-Penrose inverse, and the conjugate transpose of A, are considered. An essential role, in characterizing when such polynomials are satisfied by two matrices linked as above, is played by the condition that the column space of A is the column space of . The results given unify a number of prior, isolated results.     

Strong linear preservers of rank reverse permutability on triangular matrices

Let F be a field with ∣F∣ > 2 and T_n(F) be the set of all n × n upper triangular matrices, where n ? 2. Let k ? 2 be a given integer. A k-tuple of matrices A₁, …, A_k ∈ T_n(F) is called rank reverse permutable if rank(A₁ A₂ ? A_k) = rank(A_k A_k−1 ? A₁). We characterize the linear maps on T_n(F) that strongly preserve the set of rank reverse permutable matrix k-tuples.     

Some explicit solutions of the additive Deligne-Simpson problem and their applications

In this paper we construct three infinite series and two extra triples (E₈ and ) of complex matrices B, C, and A=B+C of special spectral types associated to Simpson's classification in Amer. Math. Soc. Proc. 1 (1992) 157 and Magyar et al. classification in Adv. Math. 141 (1999) 97. This enables us to construct Fuchsian systems of differential equations which generalize the hypergeometric equation of Gauss-Riemann. In a sense, they are the closest relatives of the famous equation, because their triples of spectral flags have finitely many orbits for the diagonal action of the general linear group in the space of solutions. In all the cases except for E₈, we also explicitly construct scalar products such that A, B, and C are self-adjoint with respect to them. In the context of Fuchsian systems, these scalar products become monodromy invariant complex symmetric bilinear forms in the spaces of solutions.When the eigenvalues of A, B, and C are real, the matrices and the scalar products become real as well. We find inequalities on the eigenvalues of A, B, and C which make the scalar products positive-definite.As proved by Klyachko, spectra of three hermitian (or real symmetric) matrices B, C, and A=B+C form a polyhedral convex cone in the space of triple spectra. He also gave a recursive algorithm to generate inequalities describing the cone. The inequalities we obtain describe non-recursively some faces of the Klyachko cone.     

Idempotency of linear combinations of three idempotent matrices, two of which are commuting

The considerations of the present paper were inspired by Baksalary [O.M. Baksalary, Idempotency of linear combinations of three idempotent matrices, two of which are disjoint, Linear Algebra Appl. 388 (2004) 67-78] who characterized all situations in which a linear combination P=c₁P₁+c₂P₂+c₃P₃, with c_i, i=1,2,3, being nonzero complex scalars and P_i, i=1,2,3, being nonzero complex idempotent matrices such that two of them, P₁ and P₂ say, are disjoint, i.e., satisfy condition P₁P₂=0=P₂P₁, is an idempotent matrix. In the present paper, by utilizing different formalism than the one used by Baksalary, the results given in the above mentioned paper are generalized by weakening the assumption expressing the disjointness of P₁ and P₂ to the commutativity condition P₁P₂=P₂P₁.     

Some lower bounds for the spectral radius of matrices using traces

Let A be an n×n matrix with eigenvalues λ₁,λ₂,…,λ_n, and let m be an integer satisfying rank(A)?m?n. If A is real, the best possible lower bound for its spectral radius in terms of m, trA and trA² is obtained. If A is any complex matrix, two lower bounds for are compared, and furthermore a new lower bound for the spectral radius is given only in terms of trA,trA²,‖A‖,‖A^∗A-AA^∗‖,n and m.     

On the projectors FF and FF

A particular version of the singular value decomposition is exploited for an extensive analysis of two orthogonal projectors, namely FF^† and F^†F, determined by a complex square matrix F and its Moore-Penrose inverse F^†. Various functions of the projectors are considered from the point of view of their nonsingularity, idempotency, nilpotency, or their relation to the known classes of matrices, such as EP, bi-EP, GP, DR, or SR. This part of the paper was inspired by Benítez and Rako?evi? [J. Benítez, V. Rako?evi?, Matrices A such that AA^† − A^†A are nonsingular, Appl. Math. Comput. 217 (2010) 3493-3503]. Further characteristics of FF^† and F^†F, with a particular attention paid on the results dealing with column and null spaces of the functions and their eigenvalues, are derived as well. Besides establishing selected exemplary results dealing with FF^† and F^†F, the paper develops a general approach whose applicability extends far beyond the characteristics provided therein.     

The hyperring of adèle classes

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We show that the theory of hyperrings, due to M. Krasner, supplies a perfect framework to understand the algebraic structure of the adèle class space HK=AK/K^× of a global field K. After promoting F₁ to a hyperfield K, we prove that a hyperring of the form R/G (where R is a ring and G⊂R^× is a subgroup of its multiplicative group) is a hyperring extension of K if and only if G∪{0} is a subfield of R. This result applies to the adèle class space which thus inherits the structure of a hyperring extension HK of K. We begin to investigate the content of an algebraic geometry over K. The category of commutative hyperring extensions of K is inclusive of: commutative algebras over fields with semi-linear homomorphisms, abelian groups with injective homomorphisms and a rather exotic land comprising homogeneous non-Desarguesian planes. Finally, we show that for a global field K of positive characteristic, the groupoid of the prime elements of the hyperring HK is canonically and equivariantly isomorphic to the groupoid of the loops of the maximal abelian cover of the curve associated to the global field K.

Video

For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=3LSKD_PfJyc.     

A contribution to perturbation analysis for total least squares problems

In this note, we present perturbation analysis for the total least squares (Tls) problems under the genericity condition. We review the three condition numbers proposed respectively by Zhou et al. (Numer. Algorithm, 51 (2009), pp. 381–399), Baboulin and Gratton (SIAM J. Matrix Anal. Appl. 32 (2011), pp. 685–699), Li and Jia (Linear Algebra Appl. 435 (2011), pp. 674–686). We also derive new perturbation bounds.     

A remark on conservative cocompletions of categories

For a set F of small categories, F-conservative cocompletions of a category are cocompletions preserving all existing colimits of type F. We prove that every category has a free F-conservative cocompletion. However, unless F is trivial, this cocompletion fails in general to be locally small.     

On the uniqueness of overcomplete dictionaries, and a practical way to retrieve them

A full-rank under-determined linear system of equations Ax = b has in general infinitely many possible solutions. In recent years there is a growing interest in the sparsest solution of this equation—the one with the fewest non-zero entries, measured by ∥x∥₀. Such solutions find applications in signal and image processing, where the topic is typically referred to as “sparse representation”. Considering the columns of A as atoms of a dictionary, it is assumed that a given signal b is a linear composition of few such atoms. Recent work established that if the desired solution x is sparse enough, uniqueness of such a result is guaranteed. Also, pursuit algorithms, approximation solvers for the above problem, are guaranteed to succeed in finding this solution.Armed with these recent results, the problem can be reversed, and formed as an implied matrix factorization problem: Given a set of vectors {b_i}, known to emerge from such sparse constructions, Ax_i = b_i, with sufficiently sparse representations x_i, we seek the matrix A. In this paper we present both theoretical and algorithmic studies of this problem. We establish the uniqueness of the dictionary A, depending on the quantity and nature of the set {b_i}, and the sparsity of {x_i}. We also describe a recently developed algorithm, the K-SVD, that practically find the matrix A, in a manner similar to the K-Means algorithm. Finally, we demonstrate this algorithm on several stylized applications in image processing.     

Two local operators and the BLUE

For a given matrix A, a matrix P such that PA = A is said to be a local identity, and such that P²A = PA is said to be a local idempotent. In the paper, some simple properties of such operators are presented. Their relation to the best linear unbiased estimation in the general Gauss-Markov model is also demonstrated.     

Quotient-reflective and bireflective subcategories of the category of preordered sets

In previous papers, the notions of “closedness” and “strong closedness” in set-based topological categories were introduced. In this paper, we give the characterization of closed and strongly closed subobjects of an object in the category Prord of preordered sets and show that they form appropriate closure operators which enjoy the basic properties like idempotency (weak) hereditariness, and productivity.We investigate the relationships between these closure operators and the well-known ones, the up- and down-closures. As a consequence, we characterize each of T₀, T₁, and T₂ preordered sets and show that each of the full subcategories of each of T₀, T₁, T₂ preordered sets is quotient-reflective in Prord. Furthermore, we give the characterization of each of pre-Hausdorff preordered sets and zero-dimensional preordered sets, and show that there is an isomorphism of the full subcategory of zero-dimensional preordered sets and the full subcategory of pre-Hausdorff preordered sets. Finally, we show that both of these subcategories are bireflective in Prord.     

Orbifolds and groupoids

We define a 2-category structure (Pre-Orb) on the category of reduced complex orbifold atlases. We construct a 2-functor F from (Pre-Orb) to the 2-category (Grp) of proper étale effective groupoid objects over the complex manifolds. Both on (Pre-Orb) and (Grp) there are natural equivalence relations on objects: (a natural extension of) equivalence of orbifold atlases on (Pre-Orb) and Morita equivalence in (Grp). We prove that F induces a bijection between the equivalence classes of its source and target.