Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
[ 0.6063218341690895 0.5289635311888953 0.5937772444966257]
[ 0.5252981913594170 0.2941669871612735 -0.798453250866314]
sage: M
[ 10.04987562112089 0 0]
[ 1.890570661398980 4.735582601355131 0]
[ 1.492555785314984 7.006153332071100 1.638930357041381]
[ 2.885607851608969 1.804330147889395 7.963520581008761]
[ 7.064764050490923 5.626248468100069 -1.197679876299471]
sage: M*G-A
[0 0 0]
[0 0 0]
[0 0 0]
[0 0 0]
[0 0 0]
sage: (G*G.transpose()-identity_matrix(3)).norm() < 10^-10
True
sage: G.row_space() == A.row_space()
True
After trac ticket #14047, the matrix can also be over the algebraic reals AA:
sage: A = matrix(AA, [[6, -8, 1],
... [4, 1, 3],
... [6, 3, 3],
... [7, 1, -5],
... [7, -3, 5]])
sage: G, M = A.gram_schmidt(orthonormal=True)
sage: G
[ 0.5970223141259934 -0.7960297521679913 0.09950371902099891]
[ 0.6063218341690895 0.5289635311888953 0.5937772444966257]
[ 0.5252981913594170 0.2941669871612735 -0.798453250866314]
sage: M
[ 10.04987562112089 0 0]
[ 1.890570661398980 4.735582601355131 0]
[ 1.492555785314984 7.006153332071100 1.638930357041381]
[ 2.885607851608969 1.804330147889395 7.963520581008761]
[ 7.064764050490923 5.626248468100069 -1.197679876299471]
Starting with complex numbers with rational real and imaginary parts. Note the use of the conjugatetranspose
when checking the orthonormality.
sage: A = matrix(QQbar, [[ -2, -I - 1, 4*I + 2, -1],
... [-4*I, -2*I + 17, 0, 9*I + 1],
... [ 1, -2*I - 6, -I + 11, -5*I + 1]])
sage: G, M = A.gram_schmidt(orthonormal=True)
sage: (M*G-A).norm() < 10^-10
True
sage: id3 = G*G.conjugate().transpose()
sage: (id3 - identity_matrix(3)).norm() < 10^-10
True
sage: G.row_space() == A.row_space() # long time
True
A square matrix with small rank. The zero vectors produced as a result of linear dependence get eliminated,
so the rows of G are a basis for the row space of A.
sage: A = matrix(QQbar, [[2, -6, 3, 8],
... [1, -3, 2, 5],
... [0, 0, 2, 4],
176 Chapter 7. Base class for matrices, part 2