We consider the related tasks of matrix completion and matrix approximation from missing data and propose adaptive sampling procedures for both problems. We show that adaptive sampling allows one to eliminate standard incoherence assumptions on the matrix row space that are necessary for passive sampling procedures. For exact recovery of a low-rank matrix, our algorithm judiciously selects a fe...

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, ...

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...

(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...

Today’s focus on sustainability within industry presents a modeling challenge that may be dealt with using dynamic programming over an infinite time horizon. However, the curse of dimensionality often results in a large number of states in these models. These large-scale models require numerically stable solution methods. The best method for infinite-horizon dynamic programming depends on both ...