Annotated Bibliography on Generalized Inverses and Applications

Nashed, M. Zuhair; Rall, L. B.

doi:10.1016/b978-0-12-514250-2.50018-4

Cited by 128 publications

(149 citation statements)

References 949 publications

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“…(See [18] for a comprehensive account of generalized inverses.) A relation between a relatively regular element and its generalized inverse is reflexive in the sense that if b is a generalized inverse of a, then a is a generalized inverse of b.…”

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confidence: 99%

A generalized Drazin inverse

1996

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1. Introduction. The main theme of this paper can be described as a study of the Drazin inverse for bounded linear operators in a Banach space X when 0 is an isolated spectral point of the operator. This inverse is useful for instance in the solution of differential equations formulated in a Banach space X. Since the elements of X rarely enter into our considerations, the exposition seems to gain in clarity when the operators are regarded as elements of the Banach algebra L(X).An element a of a complex Banach algebra A is called regular (or relatively regular) if there is x e A such that axa = a. Relatively regular elements have been extensively studied in the case that A is the Banach algebra L{X) of all bounded linear operators on a complex Banach space X\ they have been shown to generalize in certain aspects invertible operators.If a is relatively regular, then it has a generalized inverse, which is an element b e A satisfying the equations aba = a and bab = b. (See [18] for a comprehensive account of generalized inverses.) A relation between a relatively regular element and its generalized inverse is reflexive in the sense that if b is a generalized inverse of a, then a is a generalized inverse of b.In 1958, Drazin [7] introduced a different kind of a generalized inverse in associative rings and semigroups that does not have the reflexivity property but commutes with the element.

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A generalized Drazin inverse

1996

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“…The reader is referred to [12] or [18] for properties and applications of this notion. One of the properties of T † that we need is the identity T † T = P (ker T ) ⊥ .…”

Section: Proposition 24 E O Is a Strong Deformation Retract Of Ementioning

confidence: 99%

Geometry of epimorphisms and frames

Corach¹,

Pacheco

Stojanoff

2004

Proc. Amer. Math. Soc.

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Abstract. Using a bijection between the set B H of all Bessel sequences in a (separable) Hilbert space H and the space L( 2 , H) of all (bounded linear) operators from 2 to H, we endow the set F of all frames in H with a natural topology for which we determine the connected components of F . We show that each component is a homogeneous space of the group GL( 2 ) of invertible operators of 2 . This geometrical result shows that every smooth curve in F can be lifted to a curve in GL( 2 ): given a smooth curve γ in F such that γ(0) = Ξ, there exists a smooth curve Γ in GL( 2 ) such that γ = Γ · Ξ, where the dot indicates the action of GL( 2 ) over F . We also present a similar study of the set of Riesz sequences.

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“…Esta propiedad es característica de las inverses generalizadas ponderadas por un producto escalar, en el sentido de Chipman [6].…”

Section: Proposición 14 Si ∀X ∈ Imunclassified

Definición de semiproductos escalares útiles en análisis de datos

Trejos¹

2004

RMTA

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Se desarrolla la teoría necesaria para realizar el Análisis de Datos en presencia de semiproductos escalares, extendiendo los conceptos clásicos de productos escalares usualmente empleados. Para ello, retomamos las definiciones algebraicas básicas de las formas bilineales no degeneradas y vamos desarrollando todas las herramientas algebraicas necesarias. Se estudian los operadores más importantes en el espacio de individuos, como el operados V M y el operador MV . También se estudia el caso del semiproducto escalar de pesos en el espacio de variables, que en el caso de pesos nulos corresponde a la introducción de individuos suplementarios. Finalmente, llegamos a los conceptos usuales del Análisis en Componentes Principales.Palabras clave: semiproductos escalares, formas bilineales no degeneradas, semimétricas, operador de proyección ortogonal, análisis en componentes principales. AbstractWe develop the theory necessary for Data Analysis with inner semiproducts, extending tha classical concepts of inner products usually employed. For this, we use the basic algebraic definitions of non degenerated bilinear forms and develop all the algebraic tools needed. We study the most important operators on the individual space, such as the V M and the MV operators. We also study the case of the inner semiproduct of weights in the variable space, which corresponds to the introduction of supplementary individuals in the case of null weights. Finally, we arrive to the usual concepts of Principal Component Analysis.Keywords: inner semiproducts, non degenerated bilinear forms, semimetrics, orthogonal projection operator, principal component analysis. IntroducciónEm algunos métodos de Análisis de Datos, se encuentran dificultades para definir una distancia en el espacio de los individuos. Este es el caso en los análisis canónicos cuando las variables están muy correlacionadas, pues la matriz de covarianzas es entonces numéricamente no invertible, y por ello es difícil calcular la matriz de una distancia de Mahalanobis. Hemos encontrado dificultades similares para definir una distancia entre conjunciones de modalidades esplicativas [12,14,13]. Hemos por lo tanto estudiado, en una primera parte, el uso de semiproductos escalares en Análisis de Datos. En una segunda parte, estudiamos algunas propiedades -en este contexto de semiproductos escalaresde dos operadoresútiles en Análisis de Datos. Finalmente, abordamos la definición de algunos conceptos clásicos del Análisis de Datos, como los relacionados con la dispersión de una nube y el Análisis en Componentes Principales. Hacemos notar que estos temas los hemos abordado en otras publicaciones de circulación restringida [11,14,13]. Algunos resultados sobre los semiproductos escalaresSean L un espacio vectorial de dimensión finita, L * su espacio dual, H un subespacio de L y T una forma bilineal simétrica sobre L. Se denota con la misma letra T la aplicación lineal de L en L * associada a T . T /H es la restricción de la aplicación T a H. Proposición 1 Si T es una forma bilinea...

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Annotated Bibliography on Generalized Inverses and Applications

Cited by 128 publications

References 949 publications

A generalized Drazin inverse

A generalized Drazin inverse

Geometry of epimorphisms and frames

Definición de semiproductos escalares útiles en análisis de datos

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