C Elementary and special functions (search also class L5 ) C1 Integer-valued functions (e.g., factorial, binomial coefficient, permutations, combinations, floor, ceiling) C06GXFP Factorizes a positive integer n as n = n1 × n2. This routine may be used in conjunction with C06MCFP D Linear Algebra D1 Elementary vector and matrix operations D1a Elementary vector operations D1a1 Set to constant D1a...

By elementary linear algebra involving row operations (replacing row i by row i plus row j), and column exchanges, we can write G as (Ik|A) where A is some k×n−k matrix. Now the matrix H = ( −A In−k ) satisfies GH = 0. Thus every vector y = xG satisfies yH = 0. Furthermore standard linear algebra implies that the space of vectors that satisfies yH = 0 is at most k and so this must be the space ...

The row (resp. column) rank profile of a matrix describes the stair-case shape of its row (resp. column) echelon form. We here propose a new matrix invariant, the rank profile matrix, summarizing all information on the row and column rank profiles of all the leading sub-matrices. We show that this normal form exists and is unique over any ring, provided that the notion of McCoy’s rank is used, ...

We consider the nonsymmetric algebraic Riccati equation XM12X + XM11 + M22X + M21 = 0, where M11,M12,M21,M22 are real matrices of sizes n × n, n × m,m × n, m × m, respectively, and M = [Mij ]i,j=1 is an irreducible singular M matrix with zero row sums. The equation plays an important role in the study of stochastic fluid models, where the matrix −M is the generator of a Markov chain. The soluti...

The first thing taught in Math 340 is Gaussian Elimination, i.e. the process of transforming a matrix to reduced row echelon form by elementary row operations. Because this process has the effect of multiplying the matrix by an invertible matrix it has produces a new matrix for which the solution space of the corresponding linear system is unchanged. This is made precise by Theorem 2.4 below. T...

In the matrix interdiction problem, a real-valued matrix and an integer k is given. The objective is to remove a set of k matrix columns that minimizes in the residual matrix the sum of the row values, where the value of a row is defined to be the largest entry in that row. This combinatorial problem is closely related to bipartite network interdiction problem that can be applied to minimize th...

This paper investigates the reliability characteristics of a complex system having nine subsystems arranged in the form of 3x3 matrix in which each row contains three subsystems. The configuration of the row is of the type 2-out-of-3: F. Each subsystem has n units connected in series. The system fails if any one row containing three subsystems fails. The considered system analyzed incorporating...

A new computational method based on Wilson wavelets is proposed for solving a class of nonlinear stochastic It^{o}-Volterra integral equations. To do this a new stochastic operational matrix of It^{o} integration for Wilson wavelets is obtained. Block pulse functions (BPFs) and collocation method are used to generate a process to forming this matrix. Using these basis functions and their operat...

(1) A system of linear equations can be stored using an augmented matrix. The part of the matrix that does not involve the constant terms is termed the coefficient matrix. (2) Variables correspond to columns and equations correspond to rows in the coefficient matrix. The augmented matrix has an extra column corresponding to the constant terms. (3) In the paradigm where the system of linear equa...

Let P = [pij] be the probability probability transition matrix of Markov chain, so that pij is the probability that node i contacts node j. We assume that P is aperiodic, irreducible. It may or may not be symmetric. P is double–stochastic matrix, i.e. the sum of elements in each row or column is equal to 1. By the Perron–Frobenius theorem, P has a stationary distribution π = [πi] π = πP (1) whe...