Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
[ 0.e-17 0.e-16 1.000000000000000 0.e-15 0.e
[ 0.e-15 0.e-15 0.e-15 1.000000000000000 0.e
[ 0.e-12 0.e-12 0.e-12 0.e-12 1.0000000000
sage: Q*R == A
True
An example with complex numbers in QQbar, the field of algebraic numbers.
sage: A = matrix(QQbar, [[-8, 4*I + 1, -I + 2, 2*I + 1],
... [1, -2*I - 1, -I + 3, -I + 1],
... [I + 7, 2*I + 1, -2*I + 7, -I + 1],
... [I + 2, 0, I + 12, -1]])
sage: Q, R = A.QR()
sage: Q
[ -0.7302967433402215 0.2070566455055649 + 0.5383472783144687
[ 0.0912870929175277 -0.2070566455055649 - 0.3778783780476559
[ 0.6390096504226938 + 0.0912870929175277*I 0.1708217325420910 + 0.6677576817554466
[ 0.1825741858350554 + 0.0912870929175277*I -0.03623491296347385 + 0.0724698259269477
sage: R
[ 10.95445115010333 0.e-18 - 1.917028951268082*I
[ 0 4.829596256417300 + 0.e-17*I
[ 0 0
[ 0 0
sage: Q.conjugate_transpose()*Q
[1.000000000000000 + 0.e-19*I 0.e-18 + 0.e-17*I 0.e-17 + 0.e-17*
[ 0.e-18 + 0.e-17*I 1.000000000000000 + 0.e-17*I 0.e-17 + 0.e-17*
[ 0.e-17 + 0.e-17*I 0.e-17 + 0.e-17*I 1.000000000000000 + 0.e-16*
[ 0.e-16 + 0.e-16*I 0.e-16 + 0.e-16*I 0.e-16 + 0.e-16*
sage: Q*R - A
[ 0.e-17 0.e-17 + 0.e-17*I 0.e-16 + 0.e-16*I 0.e-16 + 0.e-16*I]
[ 0.e-18 0.e-17 + 0.e-17*I 0.e-16 + 0.e-16*I 0.e-15 + 0.e-15*I]
[0.e-17 + 0.e-18*I 0.e-17 + 0.e-17*I 0.e-16 + 0.e-16*I 0.e-16 + 0.e-16*I]
[0.e-18 + 0.e-18*I 0.e-18 + 0.e-17*I 0.e-16 + 0.e-16*I 0.e-15 + 0.e-16*I]
A rank-deficient rectangular matrix, with both values of the full keyword.
sage: A = matrix(QQbar, [[2, -3, 3],
... [-1, 1, -1],
... [-1, 3, -3],
... [-5, 1, -1]])
sage: Q, R = A.QR()
sage: Q
[ 0.3592106040535498 -0.5693261797050169 0.7239227659930268 0.1509015305256380]
[ -0.1796053020267749 0.1445907757980996 0 0.9730546968377341]
[ -0.1796053020267749 0.7048800320157352 0.672213996993525 -0.1378927778941174]
[ -0.8980265101338745 -0.3976246334447737 0.1551263069985058 -0.10667177157846818]
sage: R
[ 5.567764362830022 -2.694079530401624 2.694079530401624]
[ 0 3.569584777515583 -3.569584777515583]
[ 0 0 0]
[ 0 0 0]
sage: Q.conjugate_transpose()*Q
[ 1 0.e-18 0.e-18 0.e-18]
[ 0.e-18 1 0.e-18 0.e-18]
[ 0.e-18 0.e-18 1.000000000000000 0.e-18]
[ 0.e-18 0.e-18 0.e-18 1.000000000000000]
sage: Q, R = A.QR(full=False)
sage: Q
128 Chapter 7. Base class for matrices, part 2