Showing papers on "Linear map published in 1971"

Linear transformations on matrices

Abstract: Even in thi s generality , it is clear that .!l' (I , ~) is a multiplicative se migroup with an ide ntity. The invariant I can be a scalar valued fun ction, e.g. , I (X) = det (X) ; or for that matter it can describe a property, e.g., m can equal M\" (C) and I (X) can mean that X is unitary , so that we are simply asking for the s tructure of all linear transformation s T that map the unitary group into itself. Much of a beginning course in linear algebra is de voted to the study of one as pect of this question for certain choices of I; for example, if I (X) = p (X), the rank of X , then it is well known that the three standard linear operations on the rows and columns of a matrix leave p fixed and this fact permits us to compute p(X) by reducing X to some normal form. The similarity theory is another example of this problem. In this case take I (X) to be the set of all elementary divisors of the characteristic matrix of X, and then the linear operators T that we wish to study are precisely those for which I(X)=/(T(X)). In the survey paper [18, 1962] 1 some of the aspects of this general problem are disc ussed. But since the time that paper was written there have been a number of developments. The purpose of this paper is to describe some of these.

Steepest descent for singular linear operators with nonclosed range

Abstract: Let T be a bounded linear operator between two Hilbert spaces with the range of T not necessarily closed, and let T † denote the generalized inverse of T. The method of steepest descent for minimizing is shown to converge monotonically starting with x 0 = 0, to T † y for any y whose orthogonal projection on the closure of the range of T, is in the range of TT *. This set of y's is dense in the domain of T†. The method is also applied to generalized least squares solutions of a class of unbounded linear operator equations

A unified canonical form for controllable and uncontrollable linear dynamical systems

Abstract: A linear transformation is proposed which will transform an arbitrary (constant) linear dynamical system p into a certain standard (canonical) form. This particular canonical form coincides with the well-known phase-variable canonical form [l]-[5] for the case of completely controllable, scalar input, linear dynamical systems and coincides with a canonical form recently proposed by Luenberger [6] and Wonham [7] in the case of completely controllable, vector input, linear dynamical systems. For linear dynamical systems which are not completely controllable, the canonical form proposed herein displays explicitly: (i) the sub-system of p which is completely controllable and (ii) the sub-system of p which is completely uncontrollable. The explicit identification of these two sub-systems permits us to effectively implement the important fundamental stabilization theorem for constant linear dynamical systems and also a useful theorem on spectrum controllability for linear dynamical systems. An important feature...

Simplification in the computation of the sensitivity functions for constant coefficient linear systems

Abstract: It is often necessary to compute the sensitivity functions for system parameters and initial conditions in a dynamic system. In this paper, it is shown that for constant coefficient linear systems all possible sensitivity functions can be obtained by linear transformations on the solutions to (p + 2) n th-order differential equations, where p is the number of independent inputs to the system, and n is the minimal order realization for the system. The transformations required to obtain the output sensitivities for a single-output multi-input system, which is modeled in a companion form, reduce to the identity.

Conditions of finiteness on sharply 2-transitive groups

Abstract: A well known theorem of H. Zassenhaus [6], which also appears in M. Hall [1], p. 382, states that a finite sharply 2-transitive permutation group 1) is isomorphic to the group of linear transformations x--', a + m . x on a finite near-field. M ore generally, one can show that the group of linear transformations on an algebraic structure called a near-domain (see Definition A) is sharply 2-transitive and that, up to isomorphism as permutation groups, each sharply 2-transitive group is isomorphic to the group of linear transformations on a uniquely determined near-domain, [2], [3], [4]. Hence the class of sharply 2-transitive groups is completely characterized by the class of neardomains. To the authors' knowledge, the question as to the existence of near-domains which are not near-fields is open. Some results on this question are given in [4]. In this paper, a theorem is proved which states that a near-domain is a near-field if and only if a certain subset is finite. Thus the theorem of Zassenhaus which states, in the terminology used here, that every finite near-domain is a near-field, can be generalized to more relaxed conditions of finiteness (Theorem A, Coroliaries 1 and 2). The latter two results are then interpreted in terms of sharply 2-transitive groups (Corollary 3).

Multiple Eigenvalue Bifurcation for Holomorphic Mappings

Abstract: Publisher Summary This chapter discusses multiple eigenvalue bifurcation for holomorphic mappings. The existence of nontrivial solutions of the equation λu-Lu -V(u) = θ is studied in H x R , where H is a real Hilbert space; L is a bounded, compact, and symmetric linear operator acting in H; and V is a nonlinear operator, which is holomorphic in a neighborhood of θ . Thus, V has, locally, a representation as a series of continuous homogeneous operators. It is assumed that in this representation, the homogeneous operator of the lowest order V k , k> 1, is the strong gradient of a C 1 -functional l. l is not assumed to be weakly continuous. The holomorphy suggests satisfactory results for a complex space. Bifurcation has been proved with weak non-degeneracy assumptions for complex solutions. The chapter also defines the total eigenprojections corresponding to the group of eigenvalues that emanate from 0 when v is chosen in a neighborhood of the origin.

Linear transformations under which the doubly stochastic matrices are invariant

Abstract: Let [Mn(C)] denote the set of linear maps from the n X n complex matrices into themselves and let Qn denote the set of complex doubly stochastic matrices, i.e. complex matrices whose row and column sums are 1. If FE[Mn(C)] is such that F(Qn) CQn and F*(U.) CO., then there exist Ai, Bi, A, and B EE Q such that F(X) = E AiXBi + AXtJn + J,XtB (1 + m)JnXJn for all n X n complex matrices X, where Jn is the n X n matrix whose elements aie each 1/n and where the superscript t denotes transpose. m denotes the number of the Ai (or Bi). Introduction. It has been of considerable interest to study linear maps from the n X n matrices to themselves that leave certain quantities invariant [1][12]. Often these maps are necessarily of the form F(X) =AXB or AXtB with certain restrictions imposed on the n Xn matrices A and B, where the superscript t denotes transpose. For example, Marcus and Moyls [8] show that such maps which preserve spectral values are of these forms with A unimodular and B =A-'. They show in [8], [9] that such maps which preserve certain given ranks are of these forms with A and B nonsingular. Marcus and May [7] show that such maps which preserve the permanent function are of these forms with A =P,Di and B =P2D2 where the Pi are permutation matrices and the Di are diagonal matrices such that per D1D2= 1. Marcus, Minc, and Moyls [10] show that one may assume that Di = D2=I if in addition the linear map leaves the doubly stochastic matrices invariant. This paper is concerned with linear transformations which map the set of n X n generalized doubly stochastic matrices, i.e. n X n complex matrices whose row and column sums are one, into itself. It is shown that the set of such maps F which includes both F and F* is precisely the set of linear combinations of transformations of the types AXB and CXtD, where the sum of the coefficients in any such Received by the editors May 12, 1969. AMS 1968 subject classifications. Primary 1565, 1585; Secondary 1530.

Analysis and synthesis of nonlinear control systems by means of a sampled-data nonlinearity matrix

Abstract: The concept of a sampled-data nonlinearity matrix is defined. It is shown that, by means of this matrix, a direct correlation can be establihed between the nonlinearity and its equivalent gain. Thus, symmetrical single-valued and/or double-valued nonlinearities, however complicated, can be transformed directly into the respective equivalent gains. This is obtained by means of linear transformations, using numerically known matrices, i.e. matrices which are independent of the nonlinearity. Similarly, in synthesis procedures, the nonlinerities are obtained from the characteristics of the desired equivalent gain. A theorem concerning these transformations is stated and proved, and some properties of the sampled-data nonlinearity matrix are emphasised. The matrices permitting direct and reciprocal transformations are given, and two non-linearities are calculated. A stability criterion is stated, and the stability of a nonlinear system is examined. Thus, it is shown that, in the design of the system, the matrix correlations and the stability criterion allow one to obtain the edesired behaviour by a very large class of nonlinear cntrol system by acting directly on the nonlinearity.

Nonlinear and direction connections

Abstract: Nonlinear connections and direction connections are two types of connections arising in Finsler geometry. In his work on generalized sprays, P. Dazord showed that there is a relationship between these two types (nonlinear connections were called sections by him). This relationship has also been used by J. Grifone in a work on prolongation of direction connections. In this paper we examine this relationship in a general setting. In particular, we show that E. Cartan's condition "D" is necessary and sufficient for a direction connection to define a nonlinear one. Also, we prove a nonuniqueness result for direction connections associated to a given nonlinear one. 1. Connections on vector-bundles. We first recall our definition of a connection on a vector bundle [5], [6]. For a smooth (C0) vector bundle p E-*>M, set Eo = E -0 and po =p I Eo. The bundle p-'E over E is canonically isomorphic to the bundle VE of vertical vectors in the tangent bundle TE of E. Hence we have the exact sequence J p' (1) 0 p--'E--TE --p-'TM--*O of vector bundles over E, where J corresponds to the inclusion map VEC TE and p' is essentially the tangent map p*: TE--TM. A smooth nonlinear connection on the vector bundle p:E--M is a smooth splitting of (1) over Eo. Since TEj Eo = TEo and p-1EIEo =pO1E, such a splitting is given by a smooth linear map V: TEo -_+p0E (i.e. continuous linear on the fibres), satisfying the equation VJ = id. The splitting can be conveniently described by its connection map D: TEo--E defined as D = r o V, where r: p-'E--*E is the canonical surjection over p. D is continuous linear on the fibres and is smooth. The connection on p: E-M is homogeneous (resp. linear) if the map D is homogeneous of degree 1 (resp. linear) on the p* fibres of TE. For a linear connection, the splitting of (1) automatically extends to all of E; in fact, a linear connection can be defined as a splitting of (1) which is smooth over all of E. Received by the editors June 23, 1970. AMS 1969 subject classifications. Primary 5385, 5350.

An iterative procedure for the solution of linear and nonlinear equations

Abstract: The proposed procedure applies to the general linear equation $$XoA = B$$ where B denotes a given L2 function (or vector) and XoA the product of the unknown function (or vector) X by a given linear operator (or matrix) A which satisfies the inequality condition $$||UoA|| \leqslant M||U||,Mfinite$$ for any L2 function (or vector) U (∥U∥ is defined by ∥U∥=mrŪ2, with $$UV = \int\limits_a^b {U(x)V(x)dxor\sum {U_i V_i .} }$$

Linear Canonical Transformations in the Theory of Vibrational‐Rotational Interactions in Molecules. A Practical Closed Form N‐Dimensional Procedure

Abstract: A general N‐dimensional linear transformation is written, wherein each original variable xi, pi (i=1, ···, N) is expressed in closed form as a linear function of the 2N transformed variables xi′, pi′ A set of 2N2‐N relationships between the linear coefficients is then derived for the transformation to be canonical. This procedure provides one with the most general form of N‐dimensional linear canonical quantum mechanical transformation, while giving the operators in closed form. The procedure is compared with the alternate method of using unitary generators, and its advantages over that method are discussed.

Some recent applications of functional equations to geometry

Abstract: The results of applications of solutions of functional equations or of methods used in the theory of functional equations to the following subjects are discussed in this paper. Determination of all Cremona transformations which reduce linear transformations with triangular matrices to translations. One-parameter subsemigroups of affine transformations and their homomorphisms. Extensions of homomorphisms from sub-semigroups to groups generated by them. Determination of all collineations on subsets of general projective planes and their extensions to the entire plane.

Generalization of the Concept of Invariance

Abstract: Given in Rn a linear transformation represented by the square constant matrix A, and a linear subspace J ⊂ Rn, J is an A-invariant if (1.1.1) ; A J clearly idicates the transformed of J in the linear transformation A.

A least squares fit for comparison of configurations of points in two-dimensional space

Abstract: Multidimensional scaling techniques map a set of objects into geometric space, usually Euclidean. As the solutions are not unique, and linear transformations are admissible operations, two solutions for a given set of objects are not comparable owing to differences of the coordinate systems. A Transformation of coordinates to obtain a least squares fit of two configurations is derived for the two-dimensional case.

On an algebra endomorphism induced by a space map

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