Mathematical characterization of the acoustic brightness and contrast control problem in complex vector space.

2009 ◽  
Vol 125 (4) ◽  
pp. 2538-2538 ◽  
Author(s):  
Minho Song ◽  
Yang‐Hann Kim
Keyword(s):  
Vector Space ◽  
Control Problem ◽  
Complex Vector Space ◽  
Complex Vector

Linear Symmetry Classes

1976 ◽  
Vol 28 (6) ◽  
pp. 1311-1319 ◽  
Author(s):  
L. J. Cummings ◽  
R. W. Robinson
Keyword(s):  
Vector Space ◽  
Permutation Group ◽  
Cyclic Group ◽  
Symmetry Class ◽  
Complex Vector Space ◽  
Complex Vector ◽  
Linear Character ◽  
Finite Permutation Group ◽  
Symmetry Classes ◽  
Finite Dimensional

A formula is derived for the dimension of a symmetry class of tensors (over a finite dimensional complex vector space) associated with an arbitrary finite permutation group G and a linear character of x of G. This generalizes a result of the first author [3] which solved the problem in case G is a cyclic group.

scholarly journals On relationships between two linear subspaces and two orthogonal projectors

2019 ◽  
Vol 7 (1) ◽  
pp. 142-212 ◽  
Author(s):  
Yongge Tian
Keyword(s):  
Vector Space ◽  
Matrix Decomposition ◽  
Complex Vector ◽  
Linear Subspaces ◽  
Survey Paper ◽  
Finite Dimensional ◽  
Generic Position ◽  
Ep Matrix ◽  
Orthogonal Projectors

Abstract Sum and intersection of linear subspaces in a vector space over a field are fundamental operations in linear algebra. The purpose of this survey paper is to give a comprehensive approach to the sums and intersections of two linear subspaces and their orthogonal complements in the finite-dimensional complex vector space. We shall establish a variety of closed-form formulas for representing the direct sum decompositions of the m-dimensional complex column vector space 𝔺m with respect to a pair of given linear subspaces 𝒨 and 𝒩 and their operations, and use them to derive a huge amount of decomposition identities for matrix expressions composed by a pair of orthogonal projectors onto the linear subspaces. As applications, we give matrix representation for the orthogonal projectors onto the intersections of a pair of linear subspaces using various matrix decomposition identities and Moore–Penrose inverses; necessary and su˚cient conditions for two linear subspaces to be in generic position; characterization of the commutativity of a pair of orthogonal projectors; necessary and su˚cient conditions for equalities and inequalities for a pair of subspaces to hold; equalities and inequalities for norms of a pair of orthogonal projectors and their operations; as well as a collection of characterizations of EP-matrix.

scholarly journals A note on holomorphic matric automorphic factors with respect to a lattice in a complex vector space

1976 ◽  
Vol 63 ◽  
pp. 163-171 ◽  
Author(s):  
Hisasi Morikawa
Keyword(s):  
Vector Space ◽  
Complex Vector Space ◽  
Complex Vector

A holomorphic n × n-matric automorphic factor with respect to a lattice L in Cg means a system of holomorphic n × n-matrices {μα(z) | α ∈ L} such that

scholarly journals On quasi-permutation representations of finite groups

1994 ◽  
Vol 36 (3) ◽  
pp. 301-308 ◽  
Author(s):  
J. M. Burns ◽  
B. Goldsmith ◽  
B. Hartley ◽  
R. Sandling
Keyword(s):  
Finite Group ◽  
Vector Space ◽  
Permutation Group ◽  
Minimal Degree ◽  
Permutation Representation ◽  
Permutation Groups ◽  
Complex Vector Space ◽  
Complex Vector ◽  
Group Situation ◽  
A Finite Group

In [6], Wong defined a quasi-permutation group of degree n to be a finite group G of automorphisms of an n-dimensional complex vector space such that every element of G has non-negative integral trace. The terminology derives from the fact that if G is a finite group of permutations of a set ω of size n, and we think of G as acting on the complex vector space with basis ω, then the trace of an element g ∈ G is equal to the number of points of ω fixed by g. In [6] and [7], Wong studied the extent to which some facts about permutation groups generalize to the quasi-permutation group situation. Here we investigate further the analogy between permutation groups and quasipermutation groups by studying the relation between the minimal degree of a faithful permutation representation of a given finite group G and the minimal degree of a faithful quasi-permutation representation. We shall often prefer to work over the rational field rather than the complex field.

Spinors in Hilbert space

1976 ◽  
Vol 80 (2) ◽  
pp. 337-347 ◽  
Author(s):  
R. J. Plymen
Keyword(s):  
Hilbert Space ◽  
Vector Space ◽  
Orthogonal Group ◽  
Double Cover ◽  
Projective Representation ◽  
Complex Vector Space ◽  
Complex Vector ◽  
True Representation ◽  
Dimension 2 ◽  
Image Position

In 1913, É. Cartan discovered that the special orthogonal groupSO(k) has a ‘two-valued’ representation (i.e. a projective representation) on a complex vector spaceSof dimension 2n, wherek= 2nor 2n+ 1. The projective representation in question lifts to a true representation of the double cover Spin (k) ofSO(k). We restrict attention to the casek= 2n. Under the action of Spin (2n),Sbreaks up into 2 irreducible subspaces:The vectors inSare calledspinors(relative toSO(2n)), those inS+orS−are calledhalf-spinors(4).

scholarly journals Linear groups analogous to permutation groups

1963 ◽  
Vol 3 (2) ◽  
pp. 180-184 ◽  
Author(s):  
W. J. Wong
Keyword(s):  
Finite Group ◽  
Vector Space ◽  
Permutation Group ◽  
Complex Vector Space ◽  
Complex Vector ◽  
Finite Linear Group ◽  
Singular Matrices ◽  
A Finite Group ◽  
Complex Coefficients ◽  
Finite Linear

If G is a finite linear group of degree n, that is, a finite group of automorphisms of an n-dimensional complex vector space (or, equivalently, a finite group of non-singular matrices of order n with complex coefficients), I shall say that G is a quasi-permutation group if the trace of every element of G is a non-negative rational integer. The reason for this terminology is that, if G is a permutation group of degree n, its elements, considered as acting on the elements of a basis of an n-dimensional complex vector space V, induce automorphisms of V forming a group isomorphic to G. The trace of the automorphism corresponding to an element x of G is equal to the number of letters left fixed by x, and so is a non-negative integer. Thus, a permutation group of degree n has a representation as a quasi-permutation group of degree n.

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