Linear Transformations in n-Dimensional Vector Space.

1952 ◽  
Vol 59 (9) ◽  
pp. 650
Author(s):  
R. V. Kadison ◽  
H. L. Hamburger ◽  
M. E. Grimshaw
Keyword(s):  
Vector Space ◽  
Dimensional Vector ◽  
Linear Transformations ◽  
Dimensional Vector Space

scholarly journals A Norm Inequality for Linear Transformations

1961 ◽  
Vol 4 (3) ◽  
pp. 239-242
Author(s):  
B.N. Moyls ◽  
N.A. Khan
Keyword(s):  
Positive Integer ◽  
Vector Space ◽  
Hermitian Operator ◽  
Norm Inequality ◽  
Dimensional Vector ◽  
Linear Transformations ◽  
Dimensional Vector Space ◽  
Image Position ◽  
Ky Fan

In 1949 Ky Fan [1] proved the following result: Let λ1…λn be the eigenvalues of an Hermitian operator H on an n-dimensional vector space Vn. If x1, …, xq is an orthonormal set in V1, and q is a positive integer such n that 1 ≤ q ≤ n, then1

scholarly journals Baer–Levi semigroups of linear transformations

2004 ◽  
Vol 134 (3) ◽  
pp. 477-499 ◽  
Author(s):  
Suzana Mendes-Gonçalves ◽  
R. P. Sullivan
Keyword(s):  
Vector Space ◽  
Dimensional Vector ◽  
Linear Transformations ◽  
Dimensional Vector Space ◽  
Infinite Dimensional ◽  
Basic Properties ◽  
Maximal Subsemigroups ◽  
Linear Version ◽  
Infinite Set

Given an infinite-dimensional vector space V, we consider the semigroup GS (m, n) consisting of all injective linear α: V → V for which codim ran α = n, where dim V = m ≥ n ≥ ℵ0. This is a linear version of the well-known Baer–Levi semigroup BL (p, q) defined on an infinite set X, where |X| = p ≥ q ≥ ℵ0. We show that, although the basic properties of GS (m, n) are the same as those of BL (p, q), the two semigroups are never isomorphic. We also determine all left ideals of GS (m, n) and some of its maximal subsemigroups; in this, we follow previous work on BL (p, q) by Sutov and Sullivan as well as Levi and Wood.

scholarly journals The ideal structure of idempotent-generated transformation semigroups

1985 ◽  
Vol 28 (3) ◽  
pp. 319-331 ◽  
Author(s):  
M. A. Reynolds ◽  
R. P. Sullivan
Keyword(s):  
Vector Space ◽  
Dimensional Vector ◽  
Boolean Ring ◽  
Ideal Structure ◽  
Transformation Semigroups ◽  
Linear Transformations ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
The Ideal

Let X be a set and the semigroup (under composition) of all total transformations from X into itself. In ([6], Theorem 3) Howie characterised those elements of that can be written as a product of idempotents in different from the identity. We gather from review articles that his work was later extended by Evseev and Podran [3, 4] (and independently for finite X by Sullivan [15]) to the semigroup of all partial transformations of X into itself. Howie's result was generalized in a different direction by Kim [8], and it has also been considered in both a topological and a totally ordered setting (see [11] and [14] for brief summaries of this latter work). In addition, Magill [10] investigated the corresponding idea for endomorphisms of a Boolean ring, while J. A. Erdos [2] resolved the analogous problem for linear transformations of a finite–dimensional vector space.

Linear Transformations on Symmetric Spaces II

1993 ◽  
Vol 45 (2) ◽  
pp. 357-368 ◽  
Author(s):  
Ming–Huat Lim
Keyword(s):  
Vector Space ◽  
Product Space ◽  
Symmetric Spaces ◽  
Real Field ◽  
Dimensional Vector ◽  
Infinite Field ◽  
Linear Transformations ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional

AbstractLet U be a finite dimensional vector space over an infinite field F. Let U(r) denote the r–th symmetric product space over U. Let T: U(r) → U(s) be a linear transformation which sends nonzero decomposable elements to nonzero decomposable elements. Let dim U ≥ s + 1. Then we obtain the structure of T for the following cases: (I) F is algebraically closed, (II) F is the real field, and (III) T is injective.

scholarly journals Invariants of Finite Reflection Groups

1963 ◽  
Vol 22 ◽  
pp. 57-64 ◽  
Author(s):  
Louis Solomon
Keyword(s):  
Vector Space ◽  
Characteristic Zero ◽  
Polynomial Functions ◽  
Dimensional Vector ◽  
Reflection Groups ◽  
Linear Transformations ◽  
Graded Ring ◽  
Polynomial Invariants ◽  
Dimensional Vector Space ◽  
Group Of Automorphisms

Let K be a field of characteristic zero. Let V be an n-dimensional vector space over K and let S be the graded ring of polynomial functions on V. If G is a group of linear transformations of V, then G acts naturally as a group of automorphisms of S if we defineThe elements of S invariant under all γ ∈ G constitute a homogeneous subring I(S) of S called the ring of polynomial invariants of G.

scholarly journals Maximal subgroups of infinite dimensional general linear groups

1992 ◽  
Vol 53 (3) ◽  
pp. 338-351 ◽  
Author(s):  
Dugald Macpherson
Keyword(s):  
Vector Space ◽  
Maximal Subgroups ◽  
General Linear ◽  
Linear Groups ◽  
Infinite Cardinal ◽  
Dimensional Vector ◽  
Linear Transformations ◽  
Dimensional Vector Space ◽  
Infinite Dimensional ◽  
General Linear Groups

AbstractLet k be an infinite cardinal, F a field, and let GL(k, F) be the group of all non-singular linear transformations on a ki-dimensional vector space V over F. Various examples are given of maximal subgroups of GL(k, F). These include (i) stabilizers of families of subspaces of V which are like filters or ideals on a set, (ii) almost stabilizers of certain subspaces of V, (iii) almost stabilizers of a direct decomposition of V into two k-dimensional subspaces.It is also noted that GL(k, F) is not the union of any chain of length k of proper subgroups.

Anticommuting Linear Transformations

1961 ◽  
Vol 13 ◽  
pp. 614-624 ◽  
Author(s):  
H. Kestelman
Keyword(s):  
Quantum Mechanics ◽  
Vector Space ◽  
Dimensional Vector ◽  
Linear Transformations ◽  
Dimensional Vector Space ◽  
Pauli Matrices ◽  
Image Position

It is well known that any set of four anticommuting involutions (see §2) in a four-dimensional vector space can be represented by the Dirac matrices(1)where the B1,r are the Pauli matrices(2)(See (1) for a general exposition with applications to Quantum Mechanics.) One formulation, which we shall call the Dirac-Pauli theorem (2; 3; 1), isTheorem 1. If M1,M2, M3, M4 are 4 X 4 matrices satisfyingthen there is a matrix T such thatand T is unique apart from an arbitrary numerical multiplier.

Products of nilpotent linear transformations

1994 ◽  
Vol 124 (6) ◽  
pp. 1135-1150 ◽  
Author(s):  
R. P. Sullivan
Keyword(s):  
Vector Space ◽  
Dimensional Vector ◽  
Transformation Semigroups ◽  
Linear Transformations ◽  
Dimensional Vector Space ◽  
Infinite Dimensional ◽  
New Class ◽  
Free Semigroups ◽  
Linear Version

In this paper we characterise the linear transformations of an infinite-dimensional vector space that can be written as the product of nilpotent transformations. This and a linear version of Malcev's congruence on transformation semigroups are then used to construct a new class of congruence-free semigroups.

Sign in / Sign up

Export Citation Format

Share Document