Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
TESTS:
Bogus values of the eps keyword will be caught.
sage: A.singular_values(eps=’junk’)
Traceback (most recent call last):
...
ValueError: could not convert string to float: junk
REFERENCES:
AUTHOR:
•Rob Beezer - (2011-02-18)
solve_left(b)
Solve the vector equation x*A = b for a nonsingular A.
INPUT:
•self - a square matrix that is nonsigular (of full rank).
•b - a vector of the correct size. Elements of the vector must coerce into the base ring of the coefficient
matrix. In particular, if b has entries from CDF then self must have CDF as its base ring.
OUTPUT:
The unique solution x to the matrix equation x*A = b, as a vector over the same base ring as self.
ALGORITHM:
Uses the solve() routine from the SciPy scipy.linalg module, after taking the tranpose of the
coefficient matrix.
EXAMPLES:
Over the reals.
sage: A = matrix(RDF, 3,3, [1,2,5,7.6,2.3,1,1,2,-1]); A
[ 1.0 2.0 5.0]
[ 7.6 2.3 1.0]
[ 1.0 2.0 -1.0]
sage: b = vector(RDF,[1,2,3])
sage: x = A.solve_left(b); x.zero_at(1e-18) # fix noisy zeroes
(0.666666666..., 0.0, 0.333333333...)
sage: x.parent()
Vector space of dimension 3 over Real Double Field
sage: x*A
(1.0, 2.0, 3.0)
Over the complex numbers.
(-1.55765124... - 0.644483985...*I, 0.183274021... + 0.286476868...*I, 0.270818505... + 0.24
sage: A = matrix(CDF, [[ 0, -1 + 2*I, 1 - 3*I, I],
... [2 + 4*I, -2 + 3*I, -1 + 2*I, -1 - I],
... [ 2 + I, 1 - I, -1, 5],
... [ 3*I, -1 - I, -1 + I, -3 + I]])
sage: b = vector(CDF, [2 -3*I, 3, -2 + 3*I, 8])
sage: x = A.solve_left(b); x
sage: x.parent()
Vector space of dimension 4 over Complex Double Field
sage: abs(x*A - b) < 1e-14
True
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