/// <summary>
/// Gram-Schmidt orthogonalization of an m by n matrix A, such that
/// {Q, R} is returned, where A = QR, Q is m by n and orthogonal, R is
/// n by n and upper triangular matrix.
/// </summary>
/// <returns></returns>
public MMatrix[] QRGramSchmidt()
{
int m = row;
int n = col;
MMatrix Q = new MMatrix(m, n);
MMatrix R = new MMatrix(n, n);
// the first column of Q equals the first column of this matrix
for (int i = 0; i < m; ++i)
Q[i, 0] = this[i, 0];
R[0, 0] = 1;
for (int k = 0; k < n; ++k)
{
R[k, k] = Vector.L2Norm(this.Column(k));
for (int i = 0; i < m; ++i)
Q[i, k] = this[i, k] / R[k, k];
for (int j = k + 1; j < n; ++j)
{
R[k, j] = Vector.DotProduct(Q.Column(k), this.Column(j));
for (int i = 0; i < m; ++i)
this[i, j] = this[i, j] - Q[i, k] * R[k, j];
}
}
return new MMatrix[] { Q, R };
}