This information is part of the Modelica Standard Library maintained by the Modelica Association.

Syntax

           Z = Matrices.nullspace(A);
(Z, nullity) = Matrices.nullspace(A);

Description

This function calculates an orthonormal basis Z=[z_1, z_2, ...] of the nullspace of a matrix A, i.e., A*z_i=0.

The nullspace is obtained by SVD method. That is, matrix A is decomposed into the matrices S, U, V:

A = U*S*transpose(V)

with the orthonormal matrices U and V and the matrix S with

S = [S1, 0]
S1 = [diag(s); 0]

and the singular values s={s1, s2, ..., sr} of A and r=rank(A). Note, that S has the same size as A. Since U and V are orthonormal we may write

transpose(U)*A*V = [S1, 0].

Matrix S1 obviously has full column rank and therefore, the left n-r rows (n is the number of columns of A or S) of matrix V span a nullspace of A.

The nullity of matrix A is the dimension of the nullspace of A. In view of the above, it becomes clear that nullity holds

nullity = n - r

with

n = number of columns of matrix A
r = rank(A)

Example

A = [1, 2,  3, 1;
     3, 4,  5, 2;
    -1, 2, -3, 3];
(Z, nullity) = nullspace(A);

results in:

Z=[0.1715;
  -0.686;
   0.1715;
   0.686]

nullity = 1

See also

Matrices.singularValues

• 2010/05/31 by Marcus Baur, DLR-RM