Mean Value and Limit Theorems for Generalized Matrix Functions

1969 ◽  
Vol 21 ◽  
pp. 982-991 ◽  
Author(s):  
Paul J. Nikolai
Keyword(s):  
Symmetric Group ◽  
Matrix Function ◽  
Limit Theorems ◽  
Mean Value ◽  
Complex Numbers ◽  
Matrix Functions ◽  
Principal Submatrix ◽  
Generalized Matrix ◽  
Image Position ◽  
Square Matrix

Let A = [aij] denote an n-square matrix with entries in the field of complex numbers. Denote by H a subgroup of Sn, the symmetric group on the integers 1, …, n, and by a character of degree 1 on H. Thenis the generalized matrix function of A associated with H and x; e.g., if H = Sn and χ = 1, then the permanent function. If the sequences ω = (ω1, …, ωm) and ϒ = (ϒ1, …, ϒm) are m-selections, m ≦ w, of integers 1, …, n, then A [ω| ϒ] denotes the m-square generalized submatrix [aωiϒj], i, j = 1, …, m, of the n-square matrix A. If ω is an increasing m-combination, then A [ω|ω] is an m-square principal submatrix of A.

On a Combinatorial Result of R. A. Brualdi and M. Newman

1968 ◽  
Vol 20 ◽  
pp. 1056-1067 ◽  
Author(s):  
Marvin Marcus ◽  
Stephen Pierce
Keyword(s):  
Matrix Function ◽  
Generalized Matrix ◽  
Combinatorial Result ◽  
Image Position ◽  
Square Matrix

LetHbe a subgroup ofSnand letAbe ann-square matrix over a fieldƒ. Following Schur (7) we define the generalized matrix functiondH(A)by

scholarly journals Inequalities Involving the Inverses of Positive Definite Matrices

1979 ◽  
Vol 22 (1) ◽  
pp. 11-15 ◽  
Author(s):  
Russell Merris
Keyword(s):  
Symmetric Group ◽  
Permutation Group ◽  
Matrix Function ◽  
Positive Definite ◽  
Complex Character ◽  
Positive Definite Matrices ◽  
Image Position ◽  
Square Matrix

Let G be a permutation group of degree m. Let x be an irreducible complex character of G. If A = (aij) is an m-square matrix, the generalised matrix function of A based on G and x is defined byFor example if G = Sm, the full symmetric group, and x is the alternating character, then d = determinant. If G = Sm and x is identically 1, then d = permanent.

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

1977 ◽  
Vol 29 (5) ◽  
pp. 937-946
Author(s):  
Hock Ong
Keyword(s):  
Vector Space ◽  
Symmetric Group ◽  
Linear Transformation ◽  
Matrix Function ◽  
Multiplicative Group ◽  
Matrix Functions ◽  
Linear Transformations ◽  
General Matrix ◽  
The Matrix ◽  
Image Position

Let F be a field, F* be its multiplicative group and Mn(F) be the vector space of all n-square matrices over F. Let Sn be the symmetric group acting on the set {1, 2, … , n}. If G is a subgroup of Sn and λ is a function on G with values in F, then the matrix function associated with G and X, denoted by Gλ, is defined byand letℐ(G, λ) = { T : T is a linear transformation of Mn(F) to itself and Gλ(T(X)) = Gλ(X) for all X}.

Nonzero Symmetry Classes of Smallest Dimension

1980 ◽  
Vol 32 (4) ◽  
pp. 957-968 ◽  
Author(s):  
G. H. Chan ◽  
M. H. Lim
Keyword(s):  
Vector Space ◽  
Symmetric Group ◽  
Linear Mapping ◽  
Complex Numbers ◽  
Dimensional Vector ◽  
Complex Character ◽  
Tensor Power ◽  
Dimensional Vector Space ◽  
Symmetry Classes ◽  
Image Position

Let U be a k-dimensional vector space over the complex numbers. Let ⊗m U denote the mth tensor power of U where m ≧ 2. For each permutation σ in the symmetric group Sm, there exists a linear mapping P(σ) on ⊗mU such thatfor all x1, …, xm in U.Let G be a subgroup of Sm and λ an irreducible (complex) character on G. The symmetrizeris a projection of ⊗ mU. Its range is denoted by Uλm(G) or simply Uλ(G) and is called the symmetry class of tensors corresponding to G and λ.

On Derivatives and Norms of Generalized Matrix Functions and Respective Symmetric Powers

2015 ◽  
Vol 30 ◽  
Author(s):  
Sonia Carvalho ◽  
Pedro Freitas
Keyword(s):  
Symmetric Group ◽  
Symmetric Tensor ◽  
Directional Derivatives ◽  
Matrix Functions ◽  
Tensor Power ◽  
Symmetric Powers ◽  
Generalized Matrix ◽  
Derivatives Of

In recent papers, S. Carvalho and P. J. Freitas obtained formulas for directional derivatives, of all orders, of the immanant and of the m-th $\xi$-symmetric tensor power of an operator and a matrix, when $\xi$ is a character of the full symmetric group. The operator bound norm of these derivatives was also calculated. In this paper similar results are established for generalized matrix functions and for every symmetric tensor power.

Interpretation of generalized matrix functions via geometric measure of quantum entanglement

2015 ◽  
Vol 13 (07) ◽  
pp. 1550049
Author(s):  
Haixia Chang ◽  
Vehbi E. Paksoy ◽  
Fuzhen Zhang
Keyword(s):  
Representation Theory ◽  
Symmetric Group ◽  
Quantum Entanglement ◽  
Geometric Interpretation ◽  
Matrix Functions ◽  
Irreducible Characters ◽  
Geometric Measure ◽  
Generalized Matrix

By using representation theory and irreducible characters of the symmetric group, we introduce character dependent states and study their entanglement via geometric measure. We also present a geometric interpretation of generalized matrix functions via this entanglement analysis.

A Note on the van Der Waerden Permanent Conjecture

1974 ◽  
Vol 26 (02) ◽  
pp. 352-354 ◽  
Author(s):  
Jacques Dubois
Keyword(s):  
Symmetric Group ◽  
Matrix Function ◽  
Combinatorial Problems ◽  
Complex Matrix ◽  
Image Position ◽  
Square Complex ◽  
Extensive Bibliography

The permanent of an n-square complex matrix P = (pij ) is defined by where the summation extends over Sn , the symmetric group of degree n. This matrix function has considerable significance in certain combinatorial problems [6; 7]. The properties and many related problems about the permanent are presented in [3] along with an extensive bibliography.

Nonconvexity of the Generalized Numerical Range Associated with the Principal Character

2000 ◽  
Vol 43 (4) ◽  
pp. 448-458
Author(s):  
Chi-Kwong Li ◽  
Alexandru Zaharia
Keyword(s):  
Symmetric Group ◽  
Matrix Function ◽  
Numerical Range ◽  
Complex Matrix ◽  
Normal Matrices ◽  
Principal Character ◽  
Straight Line ◽  
Generalized Matrix ◽  
Generalized Numerical Range ◽  
Special Case

AbstractSuppose m and n are integers such that 1 ≤ m ≤ n. For a subgroup H of the symmetric group Sm of degree m, consider the generalized matrix function on m × m matrices B = (bij) defined by and the generalized numerical range of an n × n complex matrix A associated with dH defined byIt is known that WH(A) is convex if m = 1 or if m = n = 2. We show that there exist normal matrices A for which WH(A) is not convex if 3 ≤ m ≤ n. Moreover, for m = 2 < n, we prove that a normal matrix A with eigenvalues lying on a straight line has convex WH(A) if and only if νA is Hermitian for some nonzero ν ∈ ℂ. These results extend those of Hu, Hurley and Tam, who studied the special case when 2 ≤ m ≤ 3 ≤ n and H = Sm.

Generalized matrix functions and determinants

2014 ◽  
Vol 12 (3) ◽  
Author(s):  
Mohammad Jafari ◽  
Ali Madadi
Keyword(s):  
Characteristic Polynomial ◽  
Matrix Function ◽  
Matrix Functions ◽  
Scalar Multiple ◽  
Generalized Matrix

AbstractIn this paper we prove that, up to a scalar multiple, the determinant is the unique generalized matrix function that preserves the product or remains invariant under similarity. Also, we present a new proof for the known result that, up to a scalar multiple, the ordinary characteristic polynomial is the unique generalized characteristic polynomial for which the Cayley-Hamilton theorem remains true.

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).

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