A combinatorial method for determining the spectrum of the linear combinations of finitely many diagonalizable matrices that mutually commute

et $X_i$, $i=1,2,...,m$, be diagonalizable matrices that mutually commute. This paper provides a combinatorial method to handle the problem of when a linear combination matrix $X=sum_{i=1}^{m}c_iX_i$ is a matrix such that $sigma(X)subseteq{lambda_1, lambda_2,..., lambda_{n}}$, where $c_i$, $i=1,2,...,m$, are nonzero complex scalars and $sigma(X)$ denotes the spectrum of the matrix $X$. If the spectra of the matrices $X$ and $X_i$, $i=1,2,...,m$, are chosen as subsets of some particular sets, then this problem is equivalent to the problem of characterizing all situations in which a linear combination of some commuting special types of matrices, e.g. the matrices such that $A^k=A$, $k=2,3,...$, is also a special type of matrix. The method developed in this note makes it possible to solve such characterization problems for the linear combinations of finitely many special types of matrices. Moreover, the method is illustrated by considering the problem, which is one of the open problems left in [Linear Algebra Appl. 437 (2012) 2091-2109], of characterizing all situations in which a linear combination $X=c_1X_1+c_2X_2+c_3X_3$ is a tripotent matrix when $X_1$ is an involutory matrix and both $X_2$ and $X_3$ are tripotent matrices that mutually commute. The results obtained cover those established in the reference above

Some comments on the life and work of Jerzy K. Baksalary (1944-2005)

Following some biographical comments on Jerzy K. Baksalary (1944–2005), this article continues with personal comments by Oskar Maria Baksalary, Tadeusz Cali´nski, R.William Farebrother, Jürgen Groß, Jan Hauke, Erkki Liski, Augustyn Markiewicz, Friedrich Pukelsheim, Tarmo Pukkila, Simo Puntanen, Tomasz Szulc, Yongge Tian, Götz Trenkler, Júlia Volaufová, Haruo Yanai, and Fuzhen Zhang, on the life and work of Jerzy K. Baksalary, and with a detailed list of his publications. Our article ends with a survey by Tadeusz Cali´nski on Jerzy Baksalary’s work in block designs and a set of photographs of Jerzy Baksalary

Geometry of oblique projections

Let A be a unital C*-algebra. Denote by P the space of selfadjoint projections of A. We study the relationship between P and the spaces of projections P_a determined by the different involutions #_a induced by positive invertible elements a in A. The maps f_p: P \to P_a sending p to the unique q in P_a with the same range as p and \Omega_a: P_a \to P sending q to the unitary part of the polar decomposition of the symmetry 2q-1 are shown to be diffeomorphisms. We characterize the pairs of idempotents q, r in A with |q-r|<1 such that there exists a positive element a in A verifying that q, r are in P_a. In this case q and r can be joined by an unique short geodesic along the space of idempotents Q of A.Comment: 25 pages, Latex, to appear in Studia Mathematic

Tripotent ve grup tersinir matrislerin bazı bileşimlerinin tersinirliği

06.03.2018 tarihli ve 30352 sayılı Resmi Gazetede yayımlanan “Yükseköğretim Kanunu İle Bazı Kanun Ve Kanun Hükmünde Kararnamelerde Değişiklik Yapılması Hakkında Kanun” ile 18.06.2018 tarihli “Lisansüstü Tezlerin Elektronik Ortamda Toplanması, Düzenlenmesi ve Erişime Açılmasına İlişkin Yönerge” gereğince tam metin erişime açılmıştır.Anahtar kelimeler: Tersinirlik; İdempotent matris; Tripotent matris; Grup tersinir matris; Köşegenleştirme.Bu çalışma üç ana bölümden oluşmaktadır. Bölüm 1' de bu çalışmada bahsedilen kavramların kullanım alanları hakkında bilgi verilmektedir.Bölüm 2' de, diğer bölümler için temel teşkil edecek olan, bazı kavram, özellik ve teoremler verilmektedir.Bölüm 3' te, Bölüm 4' te kullanılacak olan grup tersinir matrisler ve tripotent matrisler hakkında temel bilgi ve teoremler verilmektedir.Bölüm 4' te, kompleks sayılar ve boyutlu tripotent matrisler olmak üzere bileşiminin tersinirliği için gerekli ve yeterli koşullar ortaya koyulmuştur. Ayrıca böyle bileşimlerin tersleri için bazı sonuçlar elde edilmektedir. Bu sonuçlardan bazıları grup tersinir matrisler için verilmektedir.Keywords: Nonsingularity; Idempotent matrix; Tripotent matrix; Group invertible matrix; Diagonalization.This study consists of three main parts. In the Chapter 1, about the application areas of the concepts discussed in this study, information are given.In the Chapter 2, being base for the other chapters, some concepts, properties, and theorems are given.In the Chapter 3, some basic informations and theorems are given about group invertible matrices and tripotent matrices, which are necessary for the Chapter 4.In the Chapter 4, it is established necessary and sufficient conditions for the nonsingularity of combinations where are tripotent matrices and are complex numbers. Moreover, it is obtained some results for the inverse of such combinations. Some of these results are given in terms of group invertible matrices

Rank Equalities Related to Generalized Inverses of Matrices and Their Applications

This paper is divided into two parts. In the first part, we develop a general method for expressing ranks of matrix expressions that involve Moore-Penrose inverses, group inverses, Drazin inverses, as well as weighted Moore-Penrose inverses of matrices. Through this method we establish a variety of valuable rank equalities related to generalized inverses of matrices mentioned above. Using them, we characterize many matrix equalities in the theory of generalized inverses of matrices and their applications. In the second part, we consider maximal and minimal possible ranks of matrix expressions that involve variant matrices, the fundamental work is concerning extreme ranks of the two linear matrix expressions $A - BXC$ and $A - B_1X_1C_1 - B_2X_2C_2$. As applications, we present a wide range of their consequences and applications in matrix theory.Comment: 245 pages, LaTe

Órdenes parciales y pre-órdenes definidos a partir de matrices inversas generalizadas

[EN] Matrix Analysis and its applications are an important area of Applied Mathematics and are the basis of many industrial applications and for engineering in general. This work can be classified as being part of Matrix Analysis. Some partial orders and pre-orders defined in terms of generalized inverses on different sets of complex matrices are studied. In the first part of this thesis the star partial order on the class of EP matrices is studied. In this work some results obtained by Merikoski and Liu are extended to the class of EP matrices. Successors and predecessors of an EP matrix are characterized, and necessary and sufficient conditions are established for they to belong to the same class. New demonstrations of some known results using the canonical form of the EP matrices are presented. In this way a result which provides decompositions for two EP matrices comparable by the star partial order, is obtained and proved. N. Castro-González, J. Vélez-Cerrada, D. S. Djordjevic, J. J. Koliha, and Y. Wei are some authors who studied spectral projectors, denoted by $A^\pi$, corresponding to the eigenvalue zero of a matrix A. For a fixed matrix A, they characterized all matrices for which their spectral projectors coincides with $A^\pi$. In this work, we restrict our attention to the class of EP matrices and spectral projectors corresponding to the eigenvalue zero are characterized. Furthermore, the projectors mentioned above are linked to star and group partial orders. The Moore-Penrose inverse appears when the approximate (in norm 2) least squares solution of an inconsistent system of linear equations is found. When norms induced by Hermitian and positive definite matrices are employed, it is necessary to use the weighted Moore-Penrose inverse. This inverse has been studied by several authors such as Y. Wei, G. Wang, S. Qiao, H. Wu. among others. In many real situations, the problem to solve is modeled by matrices having a particular structure as, for example, they are symmetric, Hermitian, normal, EP, tridiagonal, etc. In this document, the class of EP(M;N) matrices is considered, that is, EP matrices weighted with respect to two Hermitian and positive definite matrices M and N, and in that set the weighted star partial order with respect to M and N is defined. First, the square matrices that belong to this class are studied and analyzed and then for the case M = N details are provided characterizing predecessors and successors of an EP(M;M) matrix. Extending the results given by N. Matzakos y D. Pappas for weighted EP matrices, two algorithms are designed to calculate the weighted Moore-Penrose inverse of a EP(M;M) matrix. The Drazin inverse is another generalized inverse considered in this memory. A nonzero weight matrix W is considered to transform a rectangular matrix A into two square matrices, AW and WA. Then, three new pre-orders on the set of rectangular complex matrices are defined. Matrices related by these pre-orders are characterized, finding in each case a representation for them. In particular, adjacent matrices are characterized. It is also studied the class of matrices with equal weighted Drazin projectors and they are related to the new pre-order.[ES] El Análisis Matricial y sus aplicaciones constituyen un área importante de la Matemática Aplicada y son la base de muchas aplicaciones industriales y para la ingeniería en general. El presente trabajo se encuadra dentro del Análisis Matricial. Se estudian algunos órdenes parciales y pre-órdenes, definidos a partir de inversas generalizadas, sobre diferentes conjuntos de matrices complejas. En la primera parte de esta memoria se estudia el orden parcial estrella en la clase de matrices EP. En el presente trabajo se extienden algunos resultados obtenidos por Merikoski y Liu a la clase de matrices EP. Se caracterizan los sucesores y predecesores de una matriz EP dada y se establecen condiciones necesarias y suficientes para que éstos pertenezcan a la misma clase. Se presentan nuevas demostraciones de algunos resultados conocidos utilizando la forma canónica de las matrices EP. De esta manera se obtiene un teorema que proporciona descomposiciones para dos matrices EP comparables a través del orden parcial estrella. N. Castro-González, J. Vélez-Cerrada, D. S. Djordjevic, J. J. Koliha y Y. Wei son algunos de los autores que han estudiado los proyectores espectrales correspondientes al valor propio nulo de una matriz A, denotados por $A^\pi$. Para una matriz A fija, caracterizaron todas las matrices para las cuales dicho proyector coincide con $A^\pi$. En este trabajo se restringe el conjunto de estudio a la clase de matrices EP y se caracterizan los proyectores espectrales correspondientes al valor propio nulo. Más adelante, se relacionan los proyectores mencionados con los órdenes parciales estrella y grupo. La inversa de Moore-Penrose aparece cuando se busca la solución aproximada (en norma 2) por mínimos cuadrados de un sistema de ecuaciones lineales inconsistente. En los casos en que se utilizan normas inducidas por matrices hermíticas y definidas positivas, es necesario utilizar la inversa de Moore-Penrose ponderada por dichas matrices, inversa estudiada por varios autores como Y. Wei, G. Wang, S. Qiao, H. Wu, entre otros. En muchas situaciones reales, las matrices que modelizan el problema a resolver poseen una determinada estructura como, por ejemplo, el hecho de ser simétricas, hermíticas, normales, EP, tridiagonales, etc. En esta memoria se considera la clase de matrices EP(M;N), matrices EP ponderadas con respecto a dos matrices hermíticas definidas positivas M y N, y se define en ese conjunto el orden parcial estrella ponderado con respecto a M y N. Primero se estudian las matrices cuadradas que pertenecen a esta clase y luego se particulariza al caso en que M coincide con N, caracterizando los predecesores y sucesores de una matriz EP(M;M). Extendiendo los resultados de N. Matzakos y D. Pappas para matrices EP ponderadas, se diseñan dos algoritmos para calcular la inversa de Moore-Penrose ponderada de una matriz EP(M;M). La inversa de Drazin es otra de las inversas generalizadas con las que se trabaja en esta memoria. Se considera una matriz peso W no nula que transforma una matriz rectangular A en dos matrices cuadradas, AW y WA. Luego, se definen tres nuevos pre-órdenes en el conjunto de matrices rectangulares complejas. Se caracterizan las matrices que están relacionadas mediante cada uno de esos pre-órdenes, encontrando en cada caso representaciones para ellas. En particular, se caracterizan las matrices adyacentes y se estudia la clase de las matrices que tienen proyectores de Drazin ponderados iguales.[CA] L'Anàlisi Matricial i les seues aplicacions constitueixen una àrea important de la Matemàtica Aplicada i són la base de moltes aplicacions industrials i per a la enginyeria en general. Aquest treball s'enquadra doncs dins de l'Anàlisi Matricial. S'estudien alguns ordres parcials i pre-ordres, definits a partir d'inverses generalitzades, sobre diferents conjunts de matrius complexes. A la primera part d'aquesta memòria s'estudia l'ordre parcial estrella en la classe de matrius EP. En aquest treball s'estenen alguns dels resultats obtinguts per Merikoski i Liu per a la classe de matrius EP. Es caracteritzen els successors i predecessors d'una matriu EP donada i s'estableixen condicions necessàries i suficients per a que aquests pertanyin a la mateixa classe. Es presenten noves demostracions d'alguns resultats coneguts fent servir la forma canònica de les matrius EP. D'aquesta manera s'obté i es verifica un teorema que proporciona descomposicions per a dos matrius EP comparables a través de l'ordre parcial estrella. N. Castro-González, J. Vélez-Cerrada, D. S. Djordjevic, J. J. Koliha i Y. Wei són alguns dels autors que han estudiat els projectors espectrals, denotats per $A^\pi$, corresponents al valor propi nul d'una matriu A. Per a una matriu A fixa, van caracteritzar totes les matrius per a les quals aquest projector coincideix amb $A^\pi$. En aquest treball es restringeix el conjunt d'estudi a la classe de matrius EP i es caracteritzen els projectors espectrals corresponents al valor propi nul. Més endavant es relacionen els projectors mencionats amb els ordres parcials estrella i grup. La inversa de Moore-Penrose apareix quan es busca la solució aproximada (en norma 2) per mínims quadrats d'un sistema d'equacions lineals inconsistent. En els casos en que s'utilitzen normes induïdes per matrius hermítiques i definides positives, és necessari utilitzar la inversa de Moore-Penrose ponderada per aquestes matrius, inversa estudiada per varis autors com Y. Wei, G. Wang, S. Qiao, H. Wu, entre altres. En moltes situacions reals, les matrius que modelitzen el problema a resoldre posseeixen una determinada estructura com, per exemple, el fet de ser simètriques, hermítiques, normals, EP, tridiagonals, etc. En aquesta memòria es considera la classe de matrius EP(M;N), matrius EP ponderades respecte dos matrius hermítiques definides positives M i N, i es defineix en aquest conjunt l'ordre parcial estrella ponderat respecte M i N. Primer s'estudien i analitzen les matrius quadrades que pertanyen a aquesta classe i després es particularitza per al cas en que M coincideix amb N, caracteritzant els predecessors i successors d'una matriu EP(M;M). Estenent les resultats de N. Matzakos i D. Pappas per a matrius EP ponderades, es dissenyen dos algoritmes per calcular la inversa de Moore-Penrose ponderada d'una matriu EP(M;M). La inversa de Drazin és una altra de les inverses generalitzades amb les que es treballa en aquesta memòria. Es considera una matriu pes W no nul.la que transforma una matriu rectangular A en dues matrius quadrades, AW i WA. Després, es defineixen tres nous pre-ordres en el conjunt de matrius rectangulars complexes. Es caracteritzen les matrius que estan relacionades mitjançant cada un d'aquests pre-ordres, trobant en cada cas representacions per a elles. En particular, es caracteritzen les matrius adjacents i s'estudia la classe de les matrius que tenen projectors de Drazin ponderats iguals.Hernández, AE. (2016). Órdenes parciales y pre-órdenes definidos a partir de matrices inversas generalizadas [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/64070TESI

{K,s+1}- potent matrislerin bazı lineer kombinasyonları

06.03.2018 tarihli ve 30352 sayılı Resmi Gazetede yayımlanan “Yükseköğretim Kanunu İle Bazı Kanun Ve Kanun Hükmünde Kararnamelerde Değişiklik Yapılması Hakkında Kanun” ile 18.06.2018 tarihli “Lisansüstü Tezlerin Elektronik Ortamda Toplanması, Düzenlenmesi ve Erişime Açılmasına İlişkin Yönerge” gereğince tam metin erişime açılmıştır.Bu çalışmanın ilk bölümünde, konu ile ilgili literatür bilgisini içeren bir giriş verilmektedir. Çalışma, bu bölüm ile birlikte toplam dört ana bölümden oluşmaktadır. Bölüm 2'de, Bölüm 4 için temel teşkil edecek olan bazı kavram ve bazı teoremler verilmektedir. Bölüm 3'te ise bu çalışmaya esin kaynağı olan literatürde mevcut bir çalışmadaki bazı sonuçlar hatırlatılmaktadır. Bölüm 4, bu çalışmanın esas kısmını içermektedir. Bu bölümde, üç karşılıklı değişmeli {K,s+1}–potent matrisin lineer kombinasyonu {K,s+1}–potent olduğunda lineer kombinasyondaki skalerlerin neler olabileceği ile alakalı bir sonuç verilmektedir. Ayrıca, verilen K involutif matrisi için bu lineer kombinasyonu {K,s+1}–potent yapacak şekilde skalerler ve karşılıklı değişmeli {K,s+1}–potent matrisler bulan algoritmalar verilmektedir.In the first chapter it is given an introduction, which include literature information about the subject. The study consists of four main chapters with this chapter in totally. In the Chapter 2, some of the concepts and some theorems, that constitute the basis for Chapter 4, have been given. In Chapter 3, some results from the existing study in the literature have been reminded. These are the inspiration for this work. The Chapter 4 contains the original part of this work. In this Chapter, it has been established the result associated that what could be scalars in the linear combination when the linear combination of three mutually commuting matrices is {K,s+1}–potent. Moreover, several algorithms have been given for finding some scalars and some mutually commuting {K,s+1}–potent matrices such that the linear combination is {K,s+1}–potent