Sage Reference Manual: Matrices and Spaces of Matrices

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.2Full MatrixSpace of 10 by 2147483648 dense matrices over Integer Ring # 64-bitsage: MatrixSpace(ZZ,2^31,10)... # 32-bitValueError: number of rows and columns must be less than 2^31 (on a 32-bit computer -- use a 64-Full MatrixSpace of 2147483648 by 10 dense matrices over Integer Ring # 64-bitsage: MatrixSpace(ZZ,10,10).category()Category of algebras over Integer Ringsage: MatrixSpace(QQ,10).category()Category of algebras over Rational Fieldbase_extend(R)Return base extension of this matrix space to R.INPUT:•R - ringOUTPUT: a matrix spaceEXAMPLES:sage: Mat(ZZ,3,5).base_extend(QQ)Full MatrixSpace of 3 by 5 dense matrices over Rational Fieldsage: Mat(QQ,3,5).base_extend(GF(7))Traceback (most recent call last):...TypeError: no base extension definedbasis()Returns a basis for this matrix space.Warning: This will of course compute every generator of this matrix space. So for large matrices,this could take a long time, waste a massive amount of memory (for dense matrices), and is likely notvery useful. Don’t use this on large matrix spaces.EXAMPLES:sage: Mat(ZZ,2,2).basis()[[1 0] [0 1] [0 0] [0 0][0 0], [0 0], [1 0], [0 1]]cached_method(f, name=None, key=None)A decorator for cached methods.EXAMPLES:In the following examples, one can see how a cached method works in application. Below, we demonstratewhat is done behind the scenes:sage: class C:....: @cached_method....: def __hash__(self):....: print "compute hash"....: return int(5)....: @cached_method....: def f(self, x):....: print "computing cached method"....: return x*24 Chapter 1. Matrix Spaces