Linear transformations in a Boolean vector space

1966 ◽  
Vol 167 (3) ◽  
pp. 240-247 ◽  
Author(s):  
P. V. Jagannadham
Author(s):  
D. E. Rutherford

SynopsisIn the case of Boolean matrices a given eigenvector may have a variety of eigenvalues. These eigenvalues form a sublattice of the basic Boolean algebra and the structure of this sublattice is investigated. Likewise a given eigenvalue has a variety of eigenvectors which form a module of the Boolean vector space. The structure of this module is examined. It is also shown that if a vector has a unique eigenvalue λ, then λ satisfies the characteristic equation of the matrix.


1998 ◽  
Vol 57 (1) ◽  
pp. 59-71 ◽  
Author(s):  
Rachel Thomas

In this paper we consider the characterisation of those elements of a transformation semigroup S which are a product of two proper idempotents. We give a characterisation where S is the endomorphism monoid of a strong independence algebra A, and apply this to the cases where A is an arbitrary set and where A is an arbitrary vector space. The results emphasise the analogy between the idempotent generated subsemigroups of the full transformation semigroup of a set and of the semigroup of linear transformations from a vector space to itself.


10.37236/75 ◽  
2009 ◽  
Vol 16 (2) ◽  
Author(s):  
Richard P. Stanley

Promotion and evacuation are bijections on the set of linear extensions of a finite poset first defined by Schützenberger. This paper surveys the basic properties of these two operations and discusses some generalizations. Linear extensions of a finite poset $P$ may be regarded as maximal chains in the lattice $J(P)$ of order ideals of $P$. The generalizations concern permutations of the maximal chains of a wider class of posets, or more generally bijective linear transformations on the vector space with basis consisting of the maximal chains of any poset. When the poset is the lattice of subspaces of ${\Bbb F}_q^n$, then the results can be stated in terms of the expansion of certain Hecke algebra products.


1975 ◽  
Vol 27 (3) ◽  
pp. 561-572 ◽  
Author(s):  
Albert Wei

Let K be a field and Mn﹛K) denote the vector space of n X n matrices over K. Marcus [4] posed the following general problem: Let W be a subspace of Mn(K) and S a subset of W. Describe the set L(S, W) of all linear transformations T on W such that T(S) is contained in S.


1976 ◽  
Vol 28 (3) ◽  
pp. 455-472 ◽  
Author(s):  
Hock Ong ◽  
E. P. Botta

Let F be a field, Mn(F) be the vector space of all w-square matrices with entries in F and a subset of Mn(F). It is of interest to determine the structure of linear maps T : Mn(F) →Mn(F) such that . For example: Let be GL(n, C), the group of all nonsingular n X n matrices over C [5]; the subset of all rank 1 matrices in MmXn(F) [4] (MmXn(F) is the vector space of all m X n matrices over F) ; the unitary group [2] ; or the set of all matrices X in Mn(F) such that det(X) = 0 [1]. Other results in this direction can be found in [3].


2017 ◽  
Vol 103 (3) ◽  
pp. 402-419 ◽  
Author(s):  
WORACHEAD SOMMANEE ◽  
KRITSADA SANGKHANAN

Let$V$be a vector space and let$T(V)$denote the semigroup (under composition) of all linear transformations from$V$into$V$. For a fixed subspace$W$of$V$, let$T(V,W)$be the semigroup consisting of all linear transformations from$V$into$W$. In 2008, Sullivan [‘Semigroups of linear transformations with restricted range’,Bull. Aust. Math. Soc.77(3) (2008), 441–453] proved that$$\begin{eqnarray}\displaystyle Q=\{\unicode[STIX]{x1D6FC}\in T(V,W):V\unicode[STIX]{x1D6FC}\subseteq W\unicode[STIX]{x1D6FC}\} & & \displaystyle \nonumber\end{eqnarray}$$is the largest regular subsemigroup of$T(V,W)$and characterized Green’s relations on$T(V,W)$. In this paper, we determine all the maximal regular subsemigroups of$Q$when$W$is a finite-dimensional subspace of$V$over a finite field. Moreover, we compute the rank and idempotent rank of$Q$when$W$is an$n$-dimensional subspace of an$m$-dimensional vector space$V$over a finite field$F$.


1961 ◽  
Vol 4 (3) ◽  
pp. 239-242
Author(s):  
B.N. Moyls ◽  
N.A. Khan

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


1975 ◽  
Vol 27 (3) ◽  
pp. 666-678 ◽  
Author(s):  
Charalambos D. Aliprantis

W. A. J. Luxemburg and A. C. Zaanen in [7] and W. A. J. Luxemburg in [5] have studied the order properties of the order bounded linear functionals of a given Riesz space L. In this paper we consider the vector space (L, M) of the order bounded linear transformations from a given Riesz space L into a Dedekind complete Riesz space M.


1957 ◽  
Vol 11 ◽  
pp. 125-130 ◽  
Author(s):  
Takashi Ono

Let K be any field, and L a separable extension of K of finite degree. L has a structure of vector space over K, and we shall denote this space by V. The space of endomorphisms of V will be denoted by Let x be any element of L, and N(x) the norm of x relative to the extension L/K. N is then a function defined on V with values in K. We shall call N the norm form on V. The multiplicative groups of non-zero elements of K and L will be denoted by K* and L* respectively. Let H be any subgroup of if K*. Then the elements z of L* such that N(z)∈H form a subgroup of L*, which we shall denote by GH. On the other hand the elements s of such that N(sx) = Λ(s)N(x) with Λ(s)∈H for all X∈V, form obviously a subgroup of GL(V), which we shall denote by becomes an algebraic group if H=K* or {1}. In case will mean the group of linear transformations of V leaving semi-invariant the norm form of L/K and in case will mean the group of linear transformations of V leaving invariant the norm form of L/K.


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