Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
Performs Gram-Schmidt orthogonalization on the rows of the matrix, returning a new matrix and a matrix
accomplishing the transformation.
INPUT:
•self - a matrix whose rows are to be orthogonalized.
•orthonormal - default: False - if True the returned orthogonal vectors are unit vectors. This
keyword is ignored if the matrix is over RDF or CDF and the results are always orthonormal.
OUTPUT:
A pair of matrices, G and M such that if A represents self, where the parenthetical properties occur when
orthonormal = True:
•A = M*G
•The rows of G are an orthogonal (resp. orthonormal) set of vectors.
•G times the conjugate-transpose of G is a diagonal (resp. identity) matrix.
•The row space of G equals the row space of A.
•M is a full-rank matrix with zeros above the diagonal.
For exact rings, any zero vectors produced (when the original vectors are linearly dependent) are not
output, thus the orthonormal set is linearly independent, and thus a basis for the row space of the original
matrix.
Any notion of a Gram-Schmidt procedure requires that the base ring of the matrix has a fraction field
implemented. In order to arrive at an orthonormal set, it must be possible to construct square roots of
the elements of the base field. In Sage, your best option is the field of algebraic numbers, QQbar, which
properly contains the rationals and number fields.
If you have an approximate numerical matrix, then this routine requires that your base field be the real
and complex double-precision floating point numbers, RDF and CDF. In this case, the matrix is treated as
having full rank, as no attempt is made to recognize linear dependence with approximate calculations.
EXAMPLES:
Inexact Rings, Numerical Matrices:
First, the inexact rings, CDF and RDF.
sage: A = matrix(CDF, [[ 0.6454 + 0.7491*I, -0.8662 + 0.1489*I, 0.7656 - 0.00344*I],
... [-0.2913 + 0.8057*I, 0.8321 + 0.8170*I, -0.6744 + 0.9248*I],
... [ 0.2554 + 0.3517*I, -0.4454 - 0.1715*I, 0.8325 - 0.6282*I]])
sage: G, M = A.gram_schmidt()
sage: G.round(6) # random signs
[-0.422243 - 0.490087*I 0.566698 - 0.097416*I -0.500882 + 0.002251*I]
[-0.057002 - 0.495035*I -0.35059 - 0.625323*I 0.255514 - 0.415284*I]
[ 0.394105 - 0.421778*I -0.392266 - 0.039345*I -0.352905 + 0.62195*I]
sage: M.round(6) # random
[ -1.528503 0.0 0.0]
[ 0.459974 - 0.40061*I -1.741233 0.0]
[-0.934304 + 0.148868*I 0.54833 + 0.073202*I -0.550725]
sage: (A - M*G).zero_at(10^-12)
[0.0 0.0 0.0]
[0.0 0.0 0.0]
[0.0 0.0 0.0]
sage: (G*G.conjugate_transpose()).zero_at(10^-12)
[1.0 0.0 0.0]
[0.0 1.0 0.0]
[0.0 0.0 1.0]
174 Chapter 7. Base class for matrices, part 2