Linear Transformations On Matrices: The Invariance of Generalized Permutation Matrices, I

1976 ◽  
Vol 28 (3) ◽  
pp. 455-472 ◽  
Author(s):  
Hock Ong ◽  
E. P. Botta
Keyword(s):  
Vector Space ◽  
Unitary Group ◽  
Linear Transformations ◽  
Permutation Matrices ◽  
Linear Maps

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].

Linear Transformations on Matrices: The Invariance of a Class of General Matrix Functions

1967 ◽  
Vol 19 ◽  
pp. 281-290 ◽  
Author(s):  
E. P. Botta
Keyword(s):  
Vector Space ◽  
Symmetric Function ◽  
Elementary Symmetric Function ◽  
Matrix Functions ◽  
Linear Transformations ◽  
General Matrix ◽  
Linear Maps ◽  
Generalized Matrix ◽  
Image Position

Let Mm(F) be the vector space of m-square matriceswhere F is a field; let f be a function on Mm(F) to some set R. It is of interest to determine the linear maps T: Mm(F) → Mm(F) which preserve the values of the function ƒ; i.e., ƒ(T(X)) = ƒ(X) for all X. For example, if we take ƒ(X) to be the rank of X, we are asking for a determination of the types of linear operations on matrices that preserve rank. Other classical invariants that may be taken for f are the determinant, the set of eigenvalues, and the rth elementary symmetric function of the eigenvalues. Dieudonné (1), Hua (2), Jacobs (3), Marcus (4, 6, 8), Mori ta (9), and Moyls (6) have conducted extensive research in this area. A class of matrix functions that have recently aroused considerable interest (4; 7) is the generalized matrix functions in the sense of I. Schur (10).

scholarly journals Products of two idempotent transformations over arbitrary sets and vector spaces

1998 ◽  
Vol 57 (1) ◽  
pp. 59-71 ◽  
Author(s):  
Rachel Thomas
Keyword(s):  
Vector Space ◽  
Transformation Semigroup ◽  
Vector Spaces ◽  
Endomorphism Monoid ◽  
Linear Transformations ◽  
Arbitrary Vector ◽  
Full Transformation ◽  
Full Transformation Semigroup

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.

scholarly journals Promotion and Evacuation

10.37236/75 ◽  
2009 ◽  
Vol 16 (2) ◽  
Author(s):  
Richard P. Stanley
Keyword(s):  
Vector Space ◽  
Hecke Algebra ◽  
Linear Extensions ◽  
Linear Transformations ◽  
Basic Properties ◽  
Finite Poset ◽  
Order Ideals ◽  
Maximal Chains

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.

Linear Transformations Preserving the Real Orthogonal Group

1975 ◽  
Vol 27 (3) ◽  
pp. 561-572 ◽  
Author(s):  
Albert Wei
Keyword(s):  
Vector Space ◽  
General Problem ◽  
Orthogonal Group ◽  
Linear Transformations ◽  
The Real

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.

scholarly journals Multivariate Alexander colorings

2018 ◽  
Vol 27 (14) ◽  
pp. 1850076 ◽  
Author(s):  
Lorenzo Traldi
Keyword(s):  
Vector Space ◽  
Polynomial Ring ◽  
Knot Theory ◽  
Laurent Polynomial ◽  
Module Structure ◽  
Cyclic Covering ◽  
Covering Spaces ◽  
Linear Maps ◽  
Laurent Polynomial Ring ◽  
Special Case

We extend the notion of link colorings with values in an Alexander quandle to link colorings with values in a module [Formula: see text] over the Laurent polynomial ring [Formula: see text]. If [Formula: see text] is a diagram of a link [Formula: see text] with [Formula: see text] components, then the colorings of [Formula: see text] with values in [Formula: see text] form a [Formula: see text]-module [Formula: see text]. Extending a result of Inoue [Knot quandles and infinite cyclic covering spaces, Kodai Math. J. 33 (2010) 116–122], we show that [Formula: see text] is isomorphic to the module of [Formula: see text]-linear maps from the Alexander module of [Formula: see text] to [Formula: see text]. In particular, suppose [Formula: see text] is a field and [Formula: see text] is a homomorphism of rings with unity. Then [Formula: see text] defines a [Formula: see text]-module structure on [Formula: see text], which we denote [Formula: see text]. We show that the dimension of [Formula: see text] as a vector space over [Formula: see text] is determined by the images under [Formula: see text] of the elementary ideals of [Formula: see text]. This result applies in the special case of Fox tricolorings, which correspond to [Formula: see text] and [Formula: see text]. Examples show that even in this special case, the higher Alexander polynomials do not suffice to determine [Formula: see text]; this observation corrects erroneous statements of Inoue [Quandle homomorphisms of knot quandles to Alexander quandles, J. Knot Theory Ramifications 10 (2001) 813–821; op. cit.].

THE REGULAR PART OF A SEMIGROUP OF LINEAR TRANSFORMATIONS WITH RESTRICTED RANGE

2017 ◽  
Vol 103 (3) ◽  
pp. 402-419 ◽  
Author(s):  
WORACHEAD SOMMANEE ◽  
KRITSADA SANGKHANAN
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Dimensional Subspace ◽  
Finite Dimensional Subspace ◽  
Restricted Range ◽  
Linear Transformations ◽  
Dimensional Vector Space ◽  
Finite Dimensional ◽  
Idempotent Rank ◽  
Image Position

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$F$.

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

Linear Transformation on Matrices: The Invariance of a Class of General Matrix Functions. II

1968 ◽  
Vol 20 ◽  
pp. 739-748 ◽  
Author(s):  
Peter Botta
Keyword(s):  
Vector Space ◽  
Linear Transformation ◽  
Symmetric Function ◽  
Elementary Symmetric Function ◽  
Matrix Functions ◽  
General Matrix ◽  
Linear Maps

Let Mm(F) be the vector space of m-square matrices X — (Xij), i,j= 1, … , m over a field ƒ;ƒ a function on Mm(F) to some set R. It is of interest to determine the structure of the linear maps T: Mm(F) → Mm(F) that preserve the values of the function ƒ (i.e., ƒ(T(x)) — ƒ(x) for all X). For example, if we take ƒ(x) to be the rank of X, we are asking for a determination of the types of linear operations on matrices that preserve rank (6). Other classical invariants that may be taken for ƒ are the determinant, the set of eigenvalues, and the rth elementary symmetric function of the eigenvalues.

On Order Properties of Order Bounded Transformations

1975 ◽  
Vol 27 (3) ◽  
pp. 666-678 ◽  
Author(s):  
Charalambos D. Aliprantis
Keyword(s):  
Vector Space ◽  
Riesz Space ◽  
Linear Functionals ◽  
Linear Transformations ◽  
Bounded Linear

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.

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