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.

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}.

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 On induced permutation matrices and the symmetric group

1936 ◽  
Vol 5 (1) ◽  
pp. 1-13 ◽  
Author(s):  
A. C. Aitken
Keyword(s):  
Symmetric Group ◽  
Matrix Representation ◽  
Zero Element ◽  
Natural Order ◽  
Third Order ◽  
The Third ◽  
Permutation Matrices ◽  
The Matrix ◽  
Matrix Relation ◽  
Image Position

The n! operations Ai of permutations upon n different ordered symbols correspond to n! matrices Ai of the nth order, which have in each row and in each column only one non-zero element, namely a unit. Such matrices Ai are called permutation matrices, since their effect in premultiplying an arbitrary column vector x = {x1x2….xn} is to impress the permutation Ai upon the elements xi. For example the six matrices of the third orderare permutation matrices. It is convenient to denote them bywhere the bracketed indices refer to the permutations of natural order. Clearly the relation Ai Aj = Ak entails the matrix relation AiAj = Ak; in other words, the n! matrices Ai, give a matrix representation of the symmetric group of order n!.

Permanents of (0, 1)-Circulants

1964 ◽  
Vol 7 (2) ◽  
pp. 253-263 ◽  
Author(s):  
Henryk Minc
Keyword(s):  
Symmetric Group ◽  
Permutation Matrix ◽  
Image Position ◽  
Square Matrix

The permanent of an n-square matrix A = (aij) is defined bywhere the summation extends over all permutations σ of the symmetric group Sn. A matrix is said to be a (0, 1)-matrix if each of its entries is either 0 or 1. A (0, 1)-matrix of n-1 the form , where θj = 0 or 1, j = 1,…, n, and Pn is the n-square permutation matrix with ones in the (1, 2), (2, 3),…, (n-1, n), (n, 1) positions, is called a (0, 1)-circulant. Denote the (0, 1)-circulant . It has been conjectured that1

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.

An expansion for the permanent of a doubly stochastic matrix

1973 ◽  
Vol 15 (4) ◽  
pp. 504-509
Author(s):  
R. C. Griffiths
Keyword(s):  
Symmetric Group ◽  
Line Segment ◽  
Convex Set ◽  
Stochastic Matrix ◽  
Weighted Average ◽  
Stochastic Matrices ◽  
Doubly Stochastic ◽  
Survey Article ◽  
The Matrix ◽  
Image Position

The permanent of an n-square matrix A = (aij) is defined by where Sn is the symmetric group of order n. Kn will denote the convex set of all n-square doubly stochastic matrices and K0n its interior. Jn ∈ Kn will be the matrix with all elements equal to 1/n. If M ∈ K0n, then M lies on a line segment passing through Jn and another B ∈ Kn — K0n. This note gives an expansion for the permanent of such a line segment as a weighted average of permanents of matrices in Kn. For a survey article on permanents the reader is referred to Marcus and Mine [3].

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.

Trace Forms on Lie Algebras

1962 ◽  
Vol 14 ◽  
pp. 553-564 ◽  
Author(s):  
Richard Block
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Bilinear Form ◽  
Symmetric Bilinear Form ◽  
Trace Form ◽  
Finite Degree ◽  
Trace Forms ◽  
Matrix Products ◽  
The Matrix ◽  
Image Position

If L is a Lie algebra with a representation Δ a→aΔ (a in L) (of finite degree), then by the trace form f = fΔ of Δ is meant the symmetric bilinear form on L obtained by taking the trace of the matrix products:Then f is invariant, that is, f is symmetric and f(ab, c) — f(a, bc) for all a, b, c in L. By the Δ-radical L⊥ = L⊥ of L is meant the set of a in L such that f(a, b) = 0 for all b in L. Then L⊥ is an ideal and f induces a bilinear form , called a quotient trace form, on L/L⊥. Thus an algebra has a quotient trace form if and only if there exists a Lie algebra L with a representation Δ such that

Can the Number of D Electrons in Transition Metals be Measured from White Lines?

1997 ◽  
Vol 3 (S2) ◽  
pp. 957-958 ◽  
Author(s):  
P. Rez
Keyword(s):  
Transition Metals ◽  
Energy Loss ◽  
Electron Energy ◽  
Transition Elements ◽  
White Line ◽  
Line Intensities ◽  
Continuum Region ◽  
The Matrix ◽  
Image Position ◽  
Electron Energy Loss

Sharp peaks at threshold are a prominent feature of the L23 electron energy loss edges of both first and second row transition elements. Their intensity decreases monotonically as the atomic number increases across the period. It would therefore seem likely that the number of d electrons at a transition metal atom site and any variation with alloying could be measured from the L23 electron energy loss spectrum. Pearson measured the white line intensities for a series of both 3d and 4d transition metals. He normalised the white line intensity to the intensity in a continuum region 50eV wide starting 50eV above threshold. When this normalised intensity was plotted against the number of d electrons assumed for each elements he obtained a monotonie but non linear variation. The energy loss spectrum is given bywhich is a product of p<,the density of d states, and the matrix elements for transitions between 2p and d states.

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