# convex nonnegative basis vectors in nullspace of matrix

4 messages
Open this post in threaded view
|

## convex nonnegative basis vectors in nullspace of matrix

 Dear all, I want to explore the nullspace of a matrix S: I currently use the function Null from the MASS package to get a basis for the null space: > S  = matrix(nrow=3, ncol=5, c(1,0,0,-1,1,1,1,-1,-1,0,-1,0,0,0,-1)); S > MASS::Null(t(S)) My problem is that I actually need a nonnegative basis for the null space of S. There should be a unique set of convex basis vectors spanning a vector space in which each vector v satisfies sum (S %*%  v) == 0 and min(v)>=0. Is there maybe an R function that can calculate that for me? I would appreciate any help, Thanks in advance, Hannes
Open this post in threaded view
|

## Re: convex nonnegative basis vectors in nullspace of matrix

 On Wed, Apr 11, 2012 at 06:04:28AM -0700, capy_bara wrote: > Dear all, > > I want to explore the nullspace of a matrix S: I currently use the function > Null from the MASS package to get a basis for the null space: > > S  = matrix(nrow=3, ncol=5, c(1,0,0,-1,1,1,1,-1,-1,0,-1,0,0,0,-1)); S > > MASS::Null(t(S)) > My problem is that I actually need a nonnegative basis for the null space of > S. > There should be a unique set of convex basis vectors spanning a vector space > in which each vector v satisfies sum (S %*%  v) == 0 and min(v)>=0. Hi. The null space of the above matrix has dimension 2. Its intersection with nonnegative vectors in R^5 is an infinite cone. In order to restrict it to a finite set, we can consider its intersection with the set of vectors with the sum of coordinates equal to 1. Then, the solution is a finite convex polytop and we can search for its vertices. The following code searches for vertices in random directions and finds two vertices. In this simple case, the polytop is in fact a line segment, so we get its endpoints. These endpoints form a linear basis of the original null space consisting of nonnegative vectors.   library(lpSolve)   S <- matrix(nrow=3, ncol=5, c(1,0,0,-1,1,1,1,-1,-1,0,-1,0,0,0,-1))   a <- MASS::Null(t(S))   n <- nrow(a)   a1 <- rbind(a, colSums(a))   b <- rep(0, times=n+1)   b[n+1] <- 1   dir <- c(rep(">=", times=n), "==")   sol <- matrix(nrow=100, ncol=n)   for (i in seq.int(length=nrow(sol))) {       crit <- rnorm(ncol(a))       out <- lp(objective.in=crit, const.mat=a1, const.dir=dir, const.rhs=b)       sol[i, ] <- a %*% out\$solution   }   unique(round(sol, digits=10))             [,1]      [,2]      [,3]      [,4]      [,5]   [1,] 0.2500000 0.2500000 0.0000000 0.2500000 0.2500000   [2,] 0.1642631 0.3357369 0.1714738 0.1642631 0.1642631 Use this with care, since for more complex cases, this method does not guarantee that all vertices are found. So, it is not guaranteed that every nonnegative vector in the null space is a nonnegative combination of the obtained vectors. Hope this helps. Petr Savicky. ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-helpPLEASE do read the posting guide http://www.R-project.org/posting-guide.htmland provide commented, minimal, self-contained, reproducible code.