Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
•eps - default: None - the largest number which will be considered to be zero. May also be set to the
string ‘auto’. See the discussion below.
OUTPUT:
A sorted list of the singular values of the matrix, which are the diagonal entries of the “S” matrix in the
SVD decomposition. As such, the values are real and are returned as elements of RDF. The list is sorted
with larger values first, and since theory predicts these values are always positive, for a rank-deficient
matrix the list should end in zeros (but in practice may not). The length of the list is the minimum of the
row count and column count for the matrix.
The number of non-zero singular values will be the rank of the matrix. However, as a numerical matrix, it is
impossible to control the difference between zero entries and very small non-zero entries. As an informed
consumer it is up to you to use the output responsibly. We will do our best, and give you the tools to work
with the output, but we cannot give you a guarantee.
With eps set to None you will get the raw singular values and can manage them as you see fit. You
may also set eps to any positive floating point value you wish. If you set eps to ‘auto’ this routine
will compute a reasonable cutoff value, based on the size of the matrix, the largest singular value and the
smallest nonzero value representable by the 53-bit precision values used. See the discussion at page 268
of [WATKINS].
See the examples for a way to use the “verbose” facility to easily watch the zero cutoffs in action.
ALGORITHM:
The singular values come from the SVD decomposition computed by SciPy/NumPy.
EXAMPLES:
Singular values close to zero have trailing digits that may vary on different hardware. For exact matrices,
the number of non-zero singular values will equal the rank of the matrix. So for some of the doctests we
round the small singular values that ideally would be zero, to control the variability across hardware.
This matrix has a determinant of one. A chain of two or three theorems implies the product of the singular
values must also be one.
sage: A = matrix(QQ, [[ 1, 0, 0, 0, 0, 1, 3],
... [-2, 1, 1, -2, 0, -4, 0],
... [ 1, 0, 1, -4, -6, -3, 7],
... [-2, 2, 1, 1, 7, 1, -1],
... [-1, 0, -1, 5, 8, 4, -6],
... [ 4, -2, -2, 1, -3, 0, 8],
... [-2, 1, 0, 2, 7, 3, -4]])
sage: A.determinant()
1
sage: B = A.change_ring(RDF)
sage: sv = B.singular_values(); sv
[20.5239806589, 8.48683702854, 5.86168134845, 2.44291658993,
0.583197014472, 0.269332872866, 0.00255244880761]
sage: prod(sv)
1.0
An exact matrix that is obviously not of full rank, and then a computation of the singular values after
conversion to an approximate matrix.
sage: A = matrix(QQ, [[1/3, 2/3, 11/3],
... [2/3, 1/3, 7/3],
... [2/3, 5/3, 27/3]])
sage: A.rank()
2
387