/// <summary>
/// Decompose A = QR
/// </summary>
/// <param name="A">A m x n matrix, m >= n</param>
/// <param name="Q">A m x n orthogonal matrix, this is a column orthonormal matrix and has a property that Q.transpose * Q = I as the column vectors in Q are orthogonal normal vectors </param>
/// <param name="R">A n x n matrix, which is upper triangular matrix if A is invertiable</param>
public static void Factorize(IMatrix A, out IMatrix Q, out IMatrix R)
{
List <IVector> Rcols;
List <IVector> Acols = MatrixUtils.GetColumnVectors(A);
List <IVector> vstarlist = Orthogonalization.Orthogonalize(Acols, out Rcols);
List <double> norms;
List <IVector> qlist = VectorUtils.Normalize(vstarlist, out norms);
Q = MatrixUtils.GetMatrix(qlist, A.DefaultValue);
R = MatrixUtils.GetMatrix(Rcols, A.DefaultValue);
foreach (int r in R.RowKeys)
{
R[r] = R[r].Multiply(norms[r]);
}
}
public static IMatrix Invert(IMatrix A)
{
Debug.Assert(A.RowCount == A.ColCount);
int n = A.RowCount;
double one = 1;
List <IVector> AinvCols = new List <IVector>();
for (int i = 0; i < n; ++i)
{
IVector e_i = new SparseVector(n, A.DefaultValue);
e_i[i] = one;
IVector AinvCol = Solve(A, e_i);
AinvCols.Add(AinvCol);
}
IMatrix Ainv = MatrixUtils.GetMatrix(AinvCols);
return(Ainv);
}
/// <summary>
/// Another way to obtain the orthogonal basis for the null space of A by using the orthogonalization of A's columns
/// </summary>
/// <param name="A"></param>
/// <returns></returns>
public static List <IVector> FindNullSpace2(IMatrix A)
{
List <IVector> R = null;
int n = A.ColCount;
int rowCount = A.RowCount;
// column vectors of A
List <IVector> Acols = MatrixUtils.GetColumnVectors(A);
List <IVector> vstarlist = Orthogonalization.Orthogonalize(Acols, out R);
// T is a matrix with columns given by vectors in R
// T is an upper triangle
// A = (matrix with columns vstarlist) * T
IMatrix T = MatrixUtils.GetMatrix(R, A.DefaultValue);
IMatrix T_inverse = T.Transpose(); //T inverse is its transpose
// let: A = (vstarlist * T)
// then: A * T.inverse = (vstarlist * T) * T.inverse = vstarlist * (T * T.inverse) = vstarlist * I = vstarlist
// now: if vstarlist[i] is zero vector then A * T.inverse[i] = 0
// therefore conclude 1: T.inverse[i] is in null space if vstarlist[i] is zero vector, furthermore {T.inverse[i]} are independent (since T is invertible)
//
// by: nullity(A) + rank(A) = n & count({ non-zero vstarlist[i] }) = rank(A)
// therefore conclude 2: nullity(A) = n - rank(A) = n - count({ non-zero vstarlist[i] }) = count({ zero vstarlist[i] })
//
// by conclusion 1 and 2
// we have null(A) = Span({ T.inverse[i] | i is such that vstarlist[i] is zero vector})
List <IVector> result = new List <IVector>();
List <IVector> Ticols = MatrixUtils.GetColumnVectors(T_inverse);
for (int c = 0; c < vstarlist.Count; ++c)
{
if (vstarlist[c].IsEmpty)
{
result.Add(Ticols[c]);
}
}
return(result);
}
/// <summary>
/// Get the eigen values and corresponding eigen vectors for the square matrix A
/// </summary>
/// <param name="A"></param>
/// <param name="eigenValues"></param>
/// <param name="eigenVectors"></param>
public static List <IVector> FindEigenVectors(IMatrix A, out List <double> eigenValues, int K = 100, double epsilon = 1e-10)
{
Debug.Assert(A.RowCount == A.ColCount);
IMatrix T, U;
Factorize(A, out T, out U, K, epsilon);
int n = A.RowCount;
eigenValues = new List <double>();
for (int i = 0; i < n; ++i)
{
eigenValues.Add(T[i, i]);
}
List <IVector> eigenVectors = MatrixUtils.GetColumnVectors(U);
return(eigenVectors);
}