Performs Gaussian elimination on the matrix to achieve reduced row echelon form (RREF).

Parameters¶

ncols: int | None = None¶

The number of columns to perform Gaussian elimination over. The default is None which represents the number of columns of the matrix.

Returns¶

The reduced row echelon form of the input matrix.

Notes¶

The elementary row operations in Gaussian elimination are:

1. Swap the position of any two rows.

2. Multiply any row by a non-zero scalar.

3. Add any row to a scalar multiple of another row.

Examples¶

Perform Gaussian elimination to get the reduced row echelon form of \(\mathbf{A}\).

In [1]: GF = galois.GF(31)

In [2]: A = GF([[16, 12, 1, 25], [1, 10, 27, 29], [1, 0, 3, 19]]); A
Out[2]: 
GF([[16, 12,  1, 25],
    [ 1, 10, 27, 29],
    [ 1,  0,  3, 19]], order=31)

In [3]: A.row_reduce()
Out[3]: 
GF([[ 1,  0,  0, 11],
    [ 0,  1,  0,  7],
    [ 0,  0,  1, 13]], order=31)

In [4]: np.linalg.matrix_rank(A)
Out[4]: 3

Or only perform Gaussian elimination over 2 columns.

In [5]: A.row_reduce(ncols=2)
Out[5]: 
GF([[ 1,  0,  5, 14],
    [ 0,  1, 27, 17],
    [ 0,  0, 29,  5]], order=31)