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Module « numpy.linalg »

Fonction matrix_rank - module numpy.linalg

Signature de la fonction matrix_rank

def matrix_rank(A, tol=None, hermitian=False, *, rtol=None) 

Description

help(numpy.linalg.matrix_rank)

Return matrix rank of array using SVD method

Rank of the array is the number of singular values of the array that are
greater than `tol`.

Parameters
----------
A : {(M,), (..., M, N)} array_like
    Input vector or stack of matrices.
tol : (...) array_like, float, optional
    Threshold below which SVD values are considered zero. If `tol` is
    None, and ``S`` is an array with singular values for `M`, and
    ``eps`` is the epsilon value for datatype of ``S``, then `tol` is
    set to ``S.max() * max(M, N) * eps``.
hermitian : bool, optional
    If True, `A` is assumed to be Hermitian (symmetric if real-valued),
    enabling a more efficient method for finding singular values.
    Defaults to False.
rtol : (...) array_like, float, optional
    Parameter for the relative tolerance component. Only ``tol`` or
    ``rtol`` can be set at a time. Defaults to ``max(M, N) * eps``.

    .. versionadded:: 2.0.0

Returns
-------
rank : (...) array_like
    Rank of A.

Notes
-----
The default threshold to detect rank deficiency is a test on the magnitude
of the singular values of `A`.  By default, we identify singular values
less than ``S.max() * max(M, N) * eps`` as indicating rank deficiency
(with the symbols defined above). This is the algorithm MATLAB uses [1].
It also appears in *Numerical recipes* in the discussion of SVD solutions
for linear least squares [2].

This default threshold is designed to detect rank deficiency accounting
for the numerical errors of the SVD computation. Imagine that there
is a column in `A` that is an exact (in floating point) linear combination
of other columns in `A`. Computing the SVD on `A` will not produce
a singular value exactly equal to 0 in general: any difference of
the smallest SVD value from 0 will be caused by numerical imprecision
in the calculation of the SVD. Our threshold for small SVD values takes
this numerical imprecision into account, and the default threshold will
detect such numerical rank deficiency. The threshold may declare a matrix
`A` rank deficient even if the linear combination of some columns of `A`
is not exactly equal to another column of `A` but only numerically very
close to another column of `A`.

We chose our default threshold because it is in wide use. Other thresholds
are possible.  For example, elsewhere in the 2007 edition of *Numerical
recipes* there is an alternative threshold of ``S.max() *
np.finfo(A.dtype).eps / 2. * np.sqrt(m + n + 1.)``. The authors describe
this threshold as being based on "expected roundoff error" (p 71).

The thresholds above deal with floating point roundoff error in the
calculation of the SVD.  However, you may have more information about
the sources of error in `A` that would make you consider other tolerance
values to detect *effective* rank deficiency. The most useful measure
of the tolerance depends on the operations you intend to use on your
matrix. For example, if your data come from uncertain measurements with
uncertainties greater than floating point epsilon, choosing a tolerance
near that uncertainty may be preferable. The tolerance may be absolute
if the uncertainties are absolute rather than relative.

References
----------
.. [1] MATLAB reference documentation, "Rank"
       https://www.mathworks.com/help/techdoc/ref/rank.html
.. [2] W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery,
       "Numerical Recipes (3rd edition)", Cambridge University Press, 2007,
       page 795.

Examples
--------
>>> import numpy as np
>>> from numpy.linalg import matrix_rank
>>> matrix_rank(np.eye(4)) # Full rank matrix
4
>>> I=np.eye(4); I[-1,-1] = 0. # rank deficient matrix
>>> matrix_rank(I)
3
>>> matrix_rank(np.ones((4,))) # 1 dimension - rank 1 unless all 0
1
>>> matrix_rank(np.zeros((4,)))
0

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