Compute the rank of matrix A, using the singular value decomposition.
The rank is taken to be the number of singular values of A that are greater than the specified tolerance tol. If the second argument is omitted, it is taken to be
tol = max (size (A)) * sigma(1) * eps;
where eps is machine precision and sigma(1) is the largest
singular value of A.
The rank of a matrix is the number of linearly independent rows or columns
and equals the dimension of the row and column space. The function
orth may be used to compute an orthonormal basis of the column space.
For testing if a system A*x = b of linear equations
is solvable, one can use
rank (A) == rank ([A b])
In this case, x = A \ b finds a particular solution
x. The general solution is x plus the null space of matrix
A. The function null may be used to compute a basis of the
null space.
Example:
A = [1 2 3
4 5 6
7 8 9];
rank (A)
⇒ 2
In this example, the number of linearly independent rows is only 2 because the final row is a linear combination of the first two rows:
A(3,:) == -A(1,:) + 2 * A(2,:)
See also: null, orth, sprank, svd, eps.
Package: communications