Rank and nullspace of a matrix¶

The following module, rank_nullspace.py, provides the functions rank() and nullspace(). (Note that !NumPy already provides the function matrix_rank(); the function given here allows an absolute tolerance to be specified along with a relative tolerance.)

rank_nullspace.py

In [ ]:
#!python
import numpy as np
from numpy.linalg import svd

def rank(A, atol=1e-13, rtol=0):
"""Estimate the rank (i.e. the dimension of the nullspace) of a matrix.

The algorithm used by this function is based on the singular value
decomposition of A.

Parameters
----------
A : ndarray
A should be at most 2-D.  A 1-D array with length n will be treated
as a 2-D with shape (1, n)
atol : float
The absolute tolerance for a zero singular value.  Singular values
smaller than atol are considered to be zero.
rtol : float
The relative tolerance.  Singular values less than rtol*smax are
considered to be zero, where smax is the largest singular value.

If both atol and rtol are positive, the combined tolerance is the
maximum of the two; that is::
tol = max(atol, rtol * smax)
Singular values smaller than tol are considered to be zero.

Return value
------------
r : int
The estimated rank of the matrix.

--------
numpy.linalg.matrix_rank
matrix_rank is basically the same as this function, but it does not
provide the option of the absolute tolerance.
"""

A = np.atleast_2d(A)
s = svd(A, compute_uv=False)
tol = max(atol, rtol * s)
rank = int((s >= tol).sum())
return rank

def nullspace(A, atol=1e-13, rtol=0):
"""Compute an approximate basis for the nullspace of A.

The algorithm used by this function is based on the singular value
decomposition of A.

Parameters
----------
A : ndarray
A should be at most 2-D.  A 1-D array with length k will be treated
as a 2-D with shape (1, k)
atol : float
The absolute tolerance for a zero singular value.  Singular values
smaller than atol are considered to be zero.
rtol : float
The relative tolerance.  Singular values less than rtol*smax are
considered to be zero, where smax is the largest singular value.

If both atol and rtol are positive, the combined tolerance is the
maximum of the two; that is::
tol = max(atol, rtol * smax)
Singular values smaller than tol are considered to be zero.

Return value
------------
ns : ndarray
If A is an array with shape (m, k), then ns will be an array
with shape (k, n), where n is the estimated dimension of the
nullspace of A.  The columns of ns are a basis for the
nullspace; each element in numpy.dot(A, ns) will be approximately
zero.
"""

A = np.atleast_2d(A)
u, s, vh = svd(A)
tol = max(atol, rtol * s)
nnz = (s >= tol).sum()
ns = vh[nnz:].conj().T
return ns


Here's a demonstration script.

In [ ]:
#!python
import numpy as np

from rank_nullspace import rank, nullspace

def checkit(a):
print "a:"
print a
r = rank(a)
print "rank is", r
ns = nullspace(a)
print "nullspace:"
print ns
if ns.size > 0:
res = np.abs(np.dot(a, ns)).max()
print "max residual is", res

print "-"*25

a = np.array([[1.0, 2.0, 3.0], [4.0, 5.0, 6.0], [7.0, 8.0, 9.0]])
checkit(a)

print "-"*25

a = np.array([[0.0, 2.0, 3.0], [4.0, 5.0, 6.0], [7.0, 8.0, 9.0]])
checkit(a)

print "-"*25

a = np.array([[0.0, 1.0, 2.0, 4.0], [1.0, 2.0, 3.0, 4.0]])
checkit(a)

print "-"*25

a = np.array([[1.0,   1.0j,   2.0+2.0j],
[1.0j, -1.0,   -2.0+2.0j],
[0.5,   0.5j,   1.0+1.0j]])
checkit(a)

print "-"*25


And here is the output of the script.

In [ ]:
-------------------------
a:
[[ 1.  2.  3.]
[ 4.  5.  6.]
[ 7.  8.  9.]]
rank is 2
nullspace:
[[-0.40824829]
[ 0.81649658]
[-0.40824829]]
max residual is 4.4408920985e-16
-------------------------
a:
[[ 0.  2.  3.]
[ 4.  5.  6.]
[ 7.  8.  9.]]
rank is 3
nullspace:
[]
-------------------------
a:
[[ 0.  1.  2.  4.]
[ 1.  2.  3.  4.]]
rank is 2
nullspace:
[[-0.48666474 -0.61177492]
[-0.27946883  0.76717915]
[ 0.76613356 -0.15540423]
[-0.31319957 -0.11409267]]
max residual is 3.88578058619e-16
-------------------------
a:
[[ 1.0+0.j   0.0+1.j   2.0+2.j ]
[ 0.0+1.j  -1.0+0.j  -2.0+2.j ]
[ 0.5+0.j   0.0+0.5j  1.0+1.j ]]
rank is 1
nullspace:
[[ 0.00000000-0.j         -0.94868330-0.j        ]
[ 0.13333333+0.93333333j  0.00000000-0.10540926j]
[ 0.20000000-0.26666667j  0.21081851-0.21081851j]]
max residual is 1.04295984227e-15
-------------------------


Section author: WarrenWeckesser