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.

scholarly journals Some Results On The Asymptotic Behavior Of Linear Systems

1955 ◽  
Vol 7 ◽  
pp. 531-538 ◽  
Author(s):  
M. Marcus
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Matrix Differential Equation ◽  
Complex Matrix ◽  
Continuous Complex ◽  
Matrix Differential ◽  
Image Position ◽  
Vector Matrix ◽  
Complex Valued ◽  
Square Complex

1. Introduction. We consider first in §2 the asymptotic behavior as t → ∞ of the solutions of the vector-matrix differential equation(1.1) ,where A is a constant n-square complex matrix, B{t) a continuous complex valued n-square matrix defined on [0, ∞ ), and x a complex n-vector.

Some Spectral Properties of Polar Decompositions

1984 ◽  
Vol 36 (6) ◽  
pp. 973-985
Author(s):  
Bryan E. Cain
Keyword(s):  
Polar Decomposition ◽  
Positive Semidefinite ◽  
Unitary Matrix ◽  
List Type ◽  
Complex Matrix ◽  
Important Special Case ◽  
Definition Of ◽  
Image Position ◽  
Square Complex ◽  
Special Case

The results in this paper respond to two rather natural questions about a polar decomposition A = UP, where U is a unitary matrix and P is positive semidefinite. Let λ1, …, λn be the eigenvalues of A. The questions are:(A) When will |λ1|, …, |λn| be the eigenvalues of P?(B) When will λ1/|λ1|, …, λn/|λn| be the eigenvalues of U?The complete answer to (A) is “if and only if U and P commute.” In an important special case the answer to (B) is “if and only if U2 and P commute.“Since these matters are best couched in terms of two different inertias, we begin with a unifying definition of inertia which views all inertias from a single perspective.For each square complex matrix A and each complex number z let m(A, z) denote the multiplicity of z as a root of the characteristic polynomial

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.

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.

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}.

scholarly journals An Algorithm for the Permanent of Circulant Matrices

1977 ◽  
Vol 20 (1) ◽  
pp. 67-70 ◽  
Author(s):  
Larry J. Cummings ◽  
Jennifer Seberry Wallis
Keyword(s):  
Symmetric Group ◽  
Matrix Function ◽  
Permutation Matrix ◽  
Circulant Matrices ◽  
The Matrix ◽  
Image Position

The permanent of an n ✕ n matrix A = (aij) is the matrix function1where the summation is over all permutations in the symmetric group, Sn. An n ✕ n matrix A is a circulant if there are scalars a1 …, an such that2where P is the n ✕ n permutation matrix corresponding to the cycle (12 … n) in Sn.

Permanents of Random Doubly Stochastic Matrices

1974 ◽  
Vol 26 (3) ◽  
pp. 600-607 ◽  
Author(s):  
R. C. Griffiths
Keyword(s):  
Symmetric Group ◽  
Stochastic Matrix ◽  
Stochastic Matrices ◽  
Doubly Stochastic Matrix ◽  
Doubly Stochastic ◽  
Survey Article ◽  
Doubly Stochastic Matrices ◽  
Image Position

The permanent of an n × n matrix A = (aij) is defined aswhere Sn is the symmetric group of order n. For a survey article on permanents the reader is referred to [2]. An unresolved conjecture due to van der Waerden states that if A is an n × n doubly stochastic matrix; then per (A) ≧ n!/nn, with equality if and only if A = Jn = (1/n).

Frobenius Symbols and the Groups SsGL(n), O(n) and Sp(n)

1973 ◽  
Vol 25 (5) ◽  
pp. 941-959 ◽  
Author(s):  
Y. J. Abramsky ◽  
H. A. Jahn ◽  
R. C. King
Keyword(s):  
Symmetric Group ◽  
Irreducible Representations ◽  
Image Position

Frobenius [2; 3] introduced the symbolsto specify partitions and the corresponding irreducible representations of the symmetric group Ss.

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.

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