Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
[ 0.3592106040535498 -0.5693261797050169]
[-0.1796053020267749 0.1445907757980996]
[-0.1796053020267749 0.7048800320157352]
[-0.8980265101338745 -0.3976246334447737]
sage: R
[ 5.567764362830022 -2.694079530401624 2.694079530401624]
[ 0 3.569584777515583 -3.569584777515583]
sage: Q.conjugate_transpose()*Q
[ 1 0.e-18]
[0.e-18 1]
Another rank-deficient rectangular matrix, with complex entries, as a reduced decomposition.
sage: A = matrix(QQbar, [[-3*I - 3, I - 3, -12*I + 1, -2],
... [-I - 1, -2, 5*I - 1, -I - 2],
... [-4*I - 4, I - 5, -7*I, -I - 4]])
sage: Q, R = A.QR(full=False)
sage: Q
[ -0.4160251471689219 - 0.4160251471689219*I 0.5370861555295747 + 0.1790287185098583*I
[ -0.1386750490563073 - 0.1386750490563073*I
[ -0.5547001962252291 - 0.5547001962252291*I
-0.7519206177414046 - 0.2506402059138015*I
-0.2148344622118299 - 0.07161148740394329*I
sage: R
[ 7.211102550927979 3.328201177351375 - 5.269651864139676*I 7.9
[ 0 1.074172311059150 -1.6
sage: Q.conjugate_transpose()*Q
[1.000000000000000 + 0.e-18*I 0.e-18 + 0.e-18*I]
[ 0.e-17 + 0.e-17*I 1.000000000000000 + 0.e-17*I]
sage: Q*R-A
[0.e-18 + 0.e-18*I 0.e-18 + 0.e-18*I 0.e-17 + 0.e-17*I 0.e-18 + 0.e-18*I]
[0.e-18 + 0.e-18*I 0.e-18 + 0.e-18*I 0.e-18 + 0.e-17*I 0.e-18 + 0.e-18*I]
[0.e-18 + 0.e-18*I 0.e-17 + 0.e-18*I 0.e-17 + 0.e-17*I 0.e-18 + 0.e-18*I]
Results of full decompositions are cached and thus returned immutable.
sage: A = random_matrix(QQbar, 2, 2)
sage: Q, R = A.QR()
sage: Q.is_mutable()
False
sage: R.is_mutable()
False
Trivial cases return trivial results of the correct size, and we check Q itself in one case.
sage: A = zero_matrix(QQbar, 0, 10)
sage: Q, R = A.QR()
sage: Q.nrows(), Q.ncols()
(0, 0)
sage: R.nrows(), R.ncols()
(0, 10)
sage: A = zero_matrix(QQbar, 3, 0)
sage: Q, R = A.QR()
sage: Q.nrows(), Q.ncols()
(3, 3)
sage: R.nrows(), R.ncols()
(3, 0)
sage: Q
[1 0 0]
[0 1 0]
[0 0 1]
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