Null

0th

Percentile

Given a matrix, M, find a matrix N giving a basis for the (left) null space. That is crossprod(N, M) = t(N) %*% M is an all-zero matrix and N has the maximum number of linearly independent columns.

Keywords
algebra
Usage
Null(M)
Arguments
M
Input matrix. A vector is coerced to a 1-column matrix.
Details

For a basis for the (right) null space $\{x : Mx = 0\}$, use Null(t(M)).

Value

The matrix N with the basis for the (left) null space, or a matrix with zero columns if the matrix M is square and of maximal rank.

References

Venables, W. N. and Ripley, B. D. (2002) Modern Applied Statistics with S. Fourth edition. Springer.

qr, qr.Q.
library(MASS) # The function is currently defined as function(M) { tmp <- qr(M) set <- if(tmp$rank == 0L) seq_len(ncol(M)) else -seq_len(tmp$rank) qr.Q(tmp, complete = TRUE)[, set, drop = FALSE] }