Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
[0.857142857142857 0.875000000000000 0.888888888888889]
[0.900000000000000 0.909090909090909 0.916666666666667]
QR(full=True)
Returns a factorization of self as a unitary matrix and an upper-triangular matrix.
INPUT:
•full - default: True - if True then the returned matrices have dimensions as described below. If
False the R matrix has no zero rows and the columns of Q are a basis for the column space of self.
OUTPUT:
If self is an m × n matrix and full=True then this method returns a pair of matrices: Q is an m × m
unitary matrix (meaning its inverse is its conjugate-transpose) and R is an m × n upper-triangular matrix
with non-negative entries on the diagonal. For a matrix of full rank this factorization is unique (due to the
restriction to positive entries on the diagonal).
If full=False then Q has m rows and the columns form an orthonormal basis for the column space of
self. So, in particular, the conjugate-transpose of Q times Q will be an identity matrix. The matrix R
will still be upper-triangular but will also have full rank, in particular it will lack the zero rows present in
a full factorization of a rank-deficient matrix.
The results obtained when full=True are cached, hence Q and R are immutable matrices in this case.
Note: This is an exact computation, so limited to exact rings. Also the base ring needs to have a fraction
field implemented in Sage and this field must contain square roots. One example is the field of algebraic
numbers, QQbar, as used in the examples below. If you need numerical results, convert the base ring to
the field of complex double numbers, CDF, which will use a faster routine that is careful about numerical
subtleties.
ALGORITHM:
“Modified Gram-Schmidt,” Algorithm 8.1 of [TREFETHEN-BAU].
EXAMPLES:
For a nonsingular matrix, the QR decomposition is unique.
sage: A = matrix(QQbar, [[-2, 0, -4, -1, -1],
... [-2, 1, -6, -3, -1],
... [1, 1, 7, 4, 5],
... [3, 0, 8, 3, 3],
... [-1, 1, -6, -6, 5]])
sage: Q, R = A.QR()
sage: Q
[ -0.4588314677411235 -0.1260506983326509 0.3812120831224489 -0.394573711338418
[ -0.4588314677411235 0.4726901187474409 -0.05198346588033394 0.717294125164660
[ 0.2294157338705618 0.6617661662464172 0.6619227988762521 -0.180872093737548
[ 0.6882472016116853 0.1890760474989764 -0.2044682991293135 0.096630296654307
[ -0.2294157338705618 0.5357154679137663 -0.609939332995919 -0.536422031427112
sage: R
[ 4.358898943540674 -0.4588314677411235 13.07669683062202 6.194224814505168 2.982
[ 0 1.670171752907625 0.5987408170800917 -1.292019657909672 6.207
[ 0 0 5.444401659866974 5.468660610611130 -0.682
[ 0 0 0 1.027626039419836 -3.61
[ 0 0 0 0 0.02
sage: Q.conjugate_transpose()*Q
[1.000000000000000 0.e-18 0.e-17 0.e-15 0.e
[ 0.e-18 1.000000000000000 0.e-16 0.e-15 0.e
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