Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
[ 0.576980839012 - 1.17091282993*I 0.232362216253 - 0.318581071175*I 0.880672963687 -
[ -0.537261143636 - 0.686091014267*I 0.591339766401 + 0.158450627525*I -0.561877938537 +
sage: ( m - L*L.conjugate().transpose() ).norm(1) < 1e-20
True
sage: L.parent()
Full MatrixSpace of 6 by 6 dense matrices over Complex Field with 100 bits of precision
sage: L[0,0] = 0
Traceback (most recent call last):
...
ValueError: matrix is immutable; please change a copy instead (i.e., use copy(M) to change a
Here is an example that returns an incorrect answer, because the input is not positive definite:
sage: r = matrix(CDF, 2, 2, [ 1, -2*I, 2*I, 0 ]); r
[ 1.0 -2.0*I]
[ 2.0*I 0.0]
sage: r.eigenvalues()
[2.56155281281, -1.56155281281]
sage: ( r - r.conjugate().transpose() ).norm(1) < 1e-30
True
sage: L = r.cholesky_decomposition(); L
[ 1.0 0.0]
[2.0*I 2.0*I]
sage: L*L.conjugate().transpose()
[ 1.0 -2.0*I]
[ 2.0*I 8.0]
This test verifies that the caching of the two variants of the Cholesky decomposition have been cleanly
separated. It can be safely removed as part of removing this method at the end of the deprecation period.
(From trac ticket #13045.)
sage: r = matrix(CDF, 2, 2, [ 0, -2*I, 2*I, 0 ]); r
[ 0.0 -2.0*I]
[ 2.0*I 0.0]
sage: r.cholesky_decomposition()
[ 0.0 0.0]
[NaN + NaN*I NaN + NaN*I]
sage: r.cholesky()
Traceback (most recent call last):
...
ValueError: matrix is not positive definite
sage: r[0,0] = 0 # clears cache
sage: r.cholesky()
Traceback (most recent call last):
...
ValueError: matrix is not positive definite
sage: r.cholesky_decomposition()
[ 0.0 0.0]
[NaN + NaN*I NaN + NaN*I]
column_module()
Return the free module over the base ring spanned by the columns of this matrix.
EXAMPLES:
sage: t = matrix(QQ, 3, range(9)); t
[0 1 2]
[3 4 5]
[6 7 8]
141