rref

Purpose

Computes the reduced row echelon form of a matrix.

Format

y = rref(x)
Parameters:x (MxN matrix) – data
Returns:y (MxN matrix) – containing reduced row echelon form of x.

Examples

// Since (row 2) = 2*(row 1), we do not expect this
// matrix to have full rank
x   = { 1 2 3
        2 4 6
        3 5 2 };

y = rref(x);

// compute rank of x
r = sumc(sumc(abs(rref(x)')) .> 1e-15);
print "The rank of x = " r;
The rank of x = 2.000

Remarks

The tolerance used for zeroing elements is computed inside the procedure using:

tol = maxc(m|n) * eps * maxc(abs(sumc(x')));

where \(eps = 2.24e-16\).

This procedure can be used to find the rank of a matrix. It is not as stable numerically as the singular value decomposition (which is used in the rank function), but it is faster for large matrices.

There is some speed advantage in having the number of rows be greater than the number of columns, so you may want to transpose if all you care about is the rank.

The following code can be used to compute the rank of a matrix:

r = sumc(sumc(abs(y')) .> tol);

where y is the output from rref(), and tol is the tolerance used. This finds the number of rows with any nonzero elements, which gives the rank of the matrix, disregarding numeric problems.

Source

rref.src