R: Null Spaces of Matrices

Null {MASS} | R Documentation

## Null Spaces of Matrices

### Description

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.

### 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.

### 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]
    }

[Package MASS version 7.3-52 Index]