Regular stochastic matrices and digraphs

1973 ◽  
Vol 10 (1) ◽  
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.

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.

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.

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.

Genetic Code and Stochastic Matrices

2010 ◽  
pp. 91-114
Author(s):  
Sergey Petoukhov ◽  
Matthew He
Keyword(s):  
Genetic Code ◽  
Fractal Structure ◽  
Hamming Distance ◽  
Stochastic Matrix ◽  
Building Blocks ◽  
Stochastic Matrices ◽  
Frequency Distributions ◽  
Decomposition Formula ◽  
The Matrix ◽  
G 11

In this chapter, we first use the Gray code representation of the genetic code C = 00, U = 10, G = 11, and A = 01 (C pairs with G, A pairs with U) to generate a sequence of genetic code-based matrices. In connection with these code-based matrices, we use the Hamming distance to generate a sequence of numerical matrices. We then further investigate the properties of the numerical matrices and show that they are doubly stochastic and symmetric. We determine the frequency distributions of the Hamming distances, building blocks of the matrices, decomposition and iterations of matrices. We present an explicit decomposition formula for the genetic code-based matrix in terms of permutation matrices. Furthermore, we establish a relation between the genetic code and a stochastic matrix based on hydrogen bonds of DNA. Using fundamental properties of the stochastic matrices, we determine explicitly the decomposition formula of genetic code-based biperiodic table. By iterating the stochastic matrix, we demonstrate the symmetrical relations between the entries of the matrix and DNA molar concentration accumulation. The evolution matrices based on genetic code were derived by using hydrogen bondsbased symmetric stochastic (2x2)-matrices as primary building blocks. The fractal structure of the genetic code and stochastic matrices were illustrated in the process of matrix decomposition, iteration and expansion in corresponding to the fractal structure of the biperiodic table introduced by Petoukhov (2001a, 2001b, 2005).

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].

Ideal flow of Markov Chain

2018 ◽  
Vol 10 (06) ◽  
pp. 1850073
Author(s):  
Kardi Teknomo
Keyword(s):  
Transition Probability ◽  
Transition Probability Matrix ◽  
Scaling Factor ◽  
Matrix Perturbation ◽  
Flow Network ◽  
Ideal Flow ◽  
The Matrix ◽  
Strongly Connected ◽  
Flow Equilibrium ◽  
Matrix Perturbation Analysis

Ideal flow network is a strongly connected network with flow, where the flows are in steady state and conserved. The matrix of ideal flow is premagic, where vector, the sum of rows, is equal to the transposed vector containing the sum of columns. The premagic property guarantees the flow conservation in all nodes. The scaling factor as the sum of node probabilities of all nodes is equal to the total flow of an ideal flow network. The same scaling factor can also be applied to create the identical ideal flow network, which has from the same transition probability matrix. Perturbation analysis of the elements of the stationary node probability vector shows an insight that the limiting distribution or the stationary distribution is also the flow-equilibrium distribution. The process is reversible that the Markov probability matrix can be obtained from the invariant state distribution through linear algebra of ideal flow matrix. Finally, we show that recursive transformation [Formula: see text] to represent [Formula: see text]-vertices path-tracing also preserved the properties of ideal flow, which is irreducible and premagic.

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).

Characterization of convergence rates for the approximation of the stationary distribution of infinite monotone stochastic matrices

1996 ◽  
Vol 33 (04) ◽  
pp. 974-985 ◽  
Author(s):  
F. Simonot ◽  
Y. Q. Song
Keyword(s):  
Convergence Rate ◽  
Generating Function ◽  
Convergence Rates ◽  
Transition Probability ◽  
Stochastic Matrix ◽  
Input Sequence ◽  
Transition Probability Matrix ◽  
Stochastic Matrices ◽  
Preceding Result

Let P be an infinite irreducible stochastic matrix, recurrent positive and stochastically monotone and Pn be any n × n stochastic matrix with Pn ≧ Tn , where Tn denotes the n × n northwest corner truncation of P. These assumptions imply the existence of limit distributions π and π n for P and Pn respectively. We show that if the Markov chain with transition probability matrix P meets the further condition of geometric recurrence then the exact convergence rate of π n to π can be expressed in terms of the radius of convergence of the generating function of π. As an application of the preceding result, we deal with the random walk on a half line and prove that the assumption of geometric recurrence can be relaxed. We also show that if the i.i.d. input sequence (A(m)) is such that we can find a real number r 0 > 1 with , then the exact convergence rate of π n to π is characterized by r 0. Moreover, when the generating function of A is not defined for |z| > 1, we derive an upper bound for the distance between π n and π based on the moments of A.

A Note Concerning Simultaneous Integral Equations

1968 ◽  
Vol 20 ◽  
pp. 855-861 ◽  
Author(s):  
Paul Knopp ◽  
Richard Sinkhorn
Keyword(s):  
Integral Equations ◽  
Stochastic Matrix ◽  
Diagonal Matrix ◽  
Main Diagonal ◽  
Doubly Stochastic Matrix ◽  
Doubly Stochastic ◽  
Scalar Multiple

In (2) Sinkhorn showed that corresponding to each positive n × n matrix A (i.e., every aij > 0) is a unique doubly stochastic matrix of the form D1AD2, where each Dk is a diagonal matrix with a positive main diagonal. The Dk themselves are unique up to a scalar multiple. In (3) the result was extended to show that D1AD2 could be made to have arbitrarypositive row and column sums (with the reservation, of course, that the sum of the row sums equal the sum of the column sums) where A need no longer be square.

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.

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