Null Spaces of Matrices

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.

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.

See also

qr, qr.Q.

Examples

# 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]
}
#> 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]
#> }
#> <environment: 0x58be7b0>