/// <summary>
/// Gram-Schmidtian 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 ArrayMatrix[] QRGramSchmidt()
{
var m = RowCount;
var n = ColumnCount;
var A = Clone();
var Q = new ArrayMatrix(m, n);
var R = new ArrayMatrix(n, n);
// the first column of Q equals the first column of this matrix
for (var i = 1; i <= m; i++) Q[i, 1] = A[i, 1];
R[1, 1] = Complex.One;
for (var k = 1; k <= n; k++)
{
R[k, k] = new Complex(A.Column(k).Norm());
for (var i = 1; i <= m; i++) Q[i, k] = A[i, k]/R[k, k];
for (var j = k + 1; j <= n; j++)
{
R[k, j] = Dot(Q.Column(k), A.Column(j));
for (var i = 1; i <= m; i++) A[i, j] = A[i, j] - Q[i, k]*R[k, j];
}
}
return new ArrayMatrix[] {Q, R};
}
/// <summary>
/// Vertically concats matrices A and B, which do not have to be of the same height.
/// </summary>
/// <param name="A"></param>
/// <param name="B"></param>
/// <returns>Matrix [A|B]</returns>
public static ArrayMatrix VerticalConcat(ArrayMatrix A, ArrayMatrix B)
{
var C = A.Column(1);
for (var j = 2; j <= A.ColumnCount; j++) C.InsertColumn(A.Column(j), j);
for (var j = 1; j <= B.ColumnCount; j++) C.InsertColumn(B.Column(j), C.ColumnCount + 1);
return C;
}