Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
sage: R. = QQ[]
sage: S. = R.quo((b^3))
sage: A = matrix(S, [[x*y^2,2*x],[2,x^10*y]])
sage: A
[ x*y^2 2*x]
[ 2 x^10*y]
sage: A.charpoly(’T’)
T^2 + (-x^10*y - x*y^2)*T - 4*x
TESTS:
sage: P. = PolynomialRing(Rationals())
sage: u = MatrixSpace(P,3)([[0,0,a],[1,0,b],[0,1,c]])
sage: Q. = PolynomialRing(P)
sage: u.charpoly(’x’)
x^3 - c*x^2 - b*x - a
A test case from trac ticket #6442. Prior to trac ticket #12292, the call to A.det() would attempt to use
the cached charpoly, and crash if an empty dictionary was cached. We don’t cache dictionaries anymore,
but this test should still pass:
sage: z = Zp(p=5)
sage: A = matrix(z, [ [3 + O(5^1), 4 + O(5^1), 4 + O(5^1)],
... [2*5^2 + O(5^3), 2 + O(5^1), 1 + O(5^1)],
... [5 + O(5^2), 1 + O(5^1), 1 + O(5^1)] ])
sage: A.charpoly(algorithm=’hessenberg’)
Traceback (most recent call last):
...
ValueError: negative valuation
sage: A.det()
3 + O(5)
The cached polynomial should be independent of the var argument (trac ticket #12292). We check (indirectly)
that the second call uses the cached value by noting that its result is not cached:
sage: M = MatrixSpace(RR, 2)
sage: A = M(range(0, 2^2))
sage: type(A)
sage: A.charpoly(’x’)
x^2 - 3.00000000000000*x - 2.00000000000000
sage: A.charpoly(’y’)
y^2 - 3.00000000000000*y - 2.00000000000000
sage: A._cache[’charpoly’]
x^2 - 3.00000000000000*x - 2.00000000000000
AUTHORS:
•Unknown: No author specified in the file from 2009-06-25
•Sebastian Pancratz (2009-06-25): Include the division-free algorithm
cholesky()
Returns the Cholesky decomposition of a symmetric or Hermitian matrix.
INPUT:
A square matrix that is real, symmetric and positive definite. Or a square matrix that is complex, Hermitian
and positive definite. Generally, the base ring for the entries of the matrix needs to be a subfield of the
algebraic numbers (QQbar). Examples include the rational numbers (QQ), some number fields, and real
algebraic numbers and the algebraic numbers themselves.
134 Chapter 7. Base class for matrices, part 2