scholarly journals The Inequalities of Merris and Foregger for Permanents

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1782
Author(s):  
Divya K. Udayan ◽  
Kanagasabapathi Somasundaram
Keyword(s):  
Symmetric Group ◽  
Linear Algebra ◽  
Symmetric Function ◽  
Stochastic Matrix ◽  
Elementary Symmetric Function ◽  
Stochastic Matrices ◽  
Doubly Stochastic ◽  
Classes Of Matrices ◽  
The Matrix ◽  
Element Set

Conjectures on permanents are well-known unsettled conjectures in linear algebra. Let A be an n×n matrix and Sn be the symmetric group on n element set. The permanent of A is defined as perA=∑σ∈Sn∏i=1naiσ(i). The Merris conjectured that for all n×n doubly stochastic matrices (denoted by Ωn), nperA≥min1≤i≤n∑j=1nperA(j|i), where A(j|i) denotes the matrix obtained from A by deleting the jth row and ith column. Foregger raised a question whether per(tJn+(1−t)A)≤perA for 0≤t≤nn−1 and for all A∈Ωn, where Jn is a doubly stochastic matrix with each entry 1n. The Merris conjecture is one of the well-known conjectures on permanents. This conjecture is still open for n≥4. In this paper, we prove the Merris inequality for some classes of matrices. We use the sub permanent inequalities to prove our results. Foregger’s inequality is also one of the well-known inequalities on permanents, and it is not yet proved for n≥5. Using the concepts of elementary symmetric function and subpermanents, we prove the Foregger’s inequality for n=5 in [0.25, 0.6248]. Let σk(A) be the sum of all subpermanents of order k. Holens and Dokovic proposed a conjecture (Holen–Dokovic conjecture), which states that if A∈Ωn,A≠Jn and k is an integer, 1≤k≤n, then σk(A)≥(n−k+1)2nkσk−1(A). In this paper, we disprove the conjecture for n=k=4.

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].

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).

scholarly journals Diagonal Sums of Doubly Substochastic Matrices

2019 ◽  
Vol 35 ◽  
pp. 42-52
Author(s):  
Lei Cao ◽  
Zhi Chen ◽  
Xuefeng Duan ◽  
Selcuk Koyuncu ◽  
Huilan Li
Keyword(s):  
Symmetric Group ◽  
Stochastic Matrix ◽  
Convex Polytope ◽  
Stochastic Matrices ◽  
Doubly Stochastic Matrix ◽  
Doubly Stochastic ◽  
Doubly Stochastic Matrices

Let $\Omega_n$ denote the convex polytope of all $n\times n$ doubly stochastic matrices, and $\omega_{n}$ denote the convex polytope of all $n\times n$ doubly substochastic matrices. For a matrix $A\in\omega_n$, define the sub-defect of $A$ to be the smallest integer $k$ such that there exists an $(n+k)\times(n+k)$ doubly stochastic matrix containing $A$ as a submatrix. Let $\omega_{n,k}$ denote the subset of $\omega_n$ which contains all doubly substochastic matrices with sub-defect $k$. For $\pi$ a permutation of symmetric group of degree $n$, the sequence of elements $a_{1\pi(1)},a_{2\pi(2)}, \ldots, a_{n\pi(n)}$ is called the diagonal of $A$ corresponding to $\pi$. Let $h(A)$ and $l(A)$ denote the maximum and minimum diagonal sums of $A\in \omega_{n,k}$, respectively. In this paper, existing results of $h$ and $l$ functions are extended from $\Omega_n$ to $\omega_{n,k}.$ In addition, an analogue of Sylvesters law of the $h$ function on $\omega_{n,k}$ is proved.

The Term and Stochastic Ranks of a Matrix

1959 ◽  
Vol 11 ◽  
pp. 269-279 ◽  
Author(s):  
N. S. Mendelsohn ◽  
A. L. Dulmage
Keyword(s):  
Real Number ◽  
Stochastic Matrix ◽  
Real Numbers ◽  
Doubly Stochastic Matrix ◽  
Doubly Stochastic ◽  
The Matrix ◽  
Term Rank ◽  
Square Matrix

The term rank p of a matrix is the order of the largest minor which has a non-zero term in the expansion of its determinant. In a recent paper (1), the authors made the following conjecture. If S is the sum of all the entries in a square matrix of non-negative real numbers and if M is the maximum row or column sum, then the term rank p of the matrix is greater than or equal to the least integer which is greater than or equal to S/M. A generalization of this conjecture is proved in § 2.The term doubly stochastic has been used to describe a matrix of nonnegative entries in which the row and column sums are all equal to one. In this paper, by a doubly stochastic matrix, the, authors mean a matrix of non-negative entries in which the row and column sums are all equal to the same real number T.

Semilinear preservers of the immanants in the set of the doubly stochastic matrices

2017 ◽  
Vol 32 ◽  
pp. 76-97
Author(s):  
M. Antonia Duffner ◽  
Rosario Fernandes
Keyword(s):  
Symmetric Group ◽  
Real Number ◽  
Irreducible Character ◽  
Complex Field ◽  
Stochastic Matrices ◽  
Doubly Stochastic ◽  
Doubly Stochastic Matrices

Let $S_n$ denote the symmetric group of degree $n$ and $M_n$ denote the set of all $n$-by-$n$ matrices over the complex field, $\IC$. Let $\chi: S_n\rightarrow \IC$ be an irreducible character of degree greater than $1$ of $S_n$. The immanant $\dc: M_n \rightarrow \IC$ associated with $\chi$ is defined by $$ \dc(X) = \sum_{\sigma \in S_n} \chi(\sigma) \prod_{j=1}^n X_{j\sigma(j)} , \quad X = [X_{jk}] \in M_n. $$ Let $\Omega_n$ be the set of all $n$-by-$n$ doubly stochastic matrices, that is, matrices with nonnegative real entries and each row and column sum is one. We say that a map $T$ from $\Omega_n$ into $\Omega_n$ \begin{itemize} \item is semilinear if $T(\lambda S_1+(1-\lambda )S_2)=\lambda T(S_1)+(1-\lambda )T(S_2)$ for all $S_1,\ S_2\in \Omega_n$ and for all real number $\lambda$ such that $0\leq \lambda\leq 1$; \item preserves $d_{\chi }$ if $d_{\chi }(T(S))=d_{\chi }(S)$ for all $S\in\Omega_n$. \end{itemize} We characterize the semilinear surjective maps $T$ from $\Omega_n $ into $\Omega_n$ that preserve $\dc$, when the degree of $\chi$ is greater than one.

A Relationship between Arbitrary Positive Matrices and Stochastic Matrices

1966 ◽  
Vol 18 ◽  
pp. 303-306 ◽  
Author(s):  
Richard Sinkhorn
Keyword(s):  
Stochastic Matrix ◽  
Diagonal Matrix ◽  
Stochastic Matrices ◽  
Doubly Stochastic Matrix ◽  
Doubly Stochastic ◽  
Positive Matrices ◽  
Square Matrix

The author (2) has shown that corresponding to each positive square matrix A (i.e. every aij > 0) is a unique doubly stochastic matrix of the form D1AD2, where the Di are diagonal matrices with positive diagonals. This doubly stochastic matrix can be obtained as the limit of the iteration defined by alternately normalizing the rows and columns of A.In this paper, it is shown that with a sacrifice of one diagonal D it is still possible to obtain a stochastic matrix. Of course, it is necessary to modify the iteration somewhat. More precisely, it is shown that corresponding to each positive square matrix A is a unique stochastic matrix of the form DAD where D is a diagonal matrix with a positive diagonal. It is shown further how this stochastic matrix can be obtained as a limit to an iteration on A.

An Elementary Proof of Johnson–Dulmage–Mendelsohn's Refinement of Birkhoff's Theorem on Doubly Stochastic Matrices

1979 ◽  
Vol 22 (1) ◽  
pp. 81-86 ◽  
Author(s):  
Akihiro Nishi
Keyword(s):  
Upper Bound ◽  
Elementary Proof ◽  
Convex Combination ◽  
Stochastic Matrix ◽  
Stochastic Matrices ◽  
Doubly Stochastic Matrix ◽  
Doubly Stochastic ◽  
Permutation Matrices ◽  
Doubly Stochastic Matrices ◽  
Birkhoff’S Theorem

SummaryA purely combinatorial and elementary proof of Johnson-Dulmage-Mendelsohn's theorem, which gives a quite sharp upper bound on the number of permutation matrices needed for representing a doubly stochastic matrix by their convex combination, is given.

Regular stochastic matrices and digraphs

1973 ◽  
Vol 10 (01) ◽  
pp. 241-243
Author(s):  
Elizabeth Berman
Keyword(s):  
Stochastic Matrix ◽  
Main Diagonal ◽  
Stochastic Matrices ◽  
The Matrix ◽  
Strongly Connected ◽  
Positive Entry

This paper presents an algorithm to determine whether a stochastic matrix is regular. The main theorem is the following. Hypothesis: An n-by-n stochastic matrix has at least one positive entry off the main diagonal in every row and column. There is at most one row with n — 1 zeros and at most one column with n — 1 zeros. There are no j-by-k submatrices consisting entirely of zeros, where j and k are integers greater than 1, with j + k = n. Conclusion: The matrix is regular. Similar results hold for strongly connected digraphs.

Log-Concavity and the Spectral Gap of Stochastic Matrices

1997 ◽  
Vol 6 (3) ◽  
pp. 371-379
Author(s):  
EKKEHARD WEINECK
Keyword(s):  
Characteristic Polynomial ◽  
Spectral Gap ◽  
Stochastic Matrix ◽  
Identity Matrix ◽  
Combinatorial Approach ◽  
Stochastic Matrices ◽  
Largest Eigenvalue ◽  
The Matrix ◽  
Second Largest Eigenvalue

Let Q be a stochastic matrix and I be the identity matrix. We show by a direct combinatorial approach that the coefficients of the characteristic polynomial of the matrix I−Q are log-concave. We use this fact to prove a new bound for the second-largest eigenvalue of Q.

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