Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

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Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 [0.750000000000000 0.800000000000000 0.833333333333333] [0.857142857142857 0.875000000000000 0.888888888888889] [0.900000000000000 0.909090909090909 0.916666666666667] sage: matrix(SR, 2, 2, range(4)).n() [0.000000000000000 1.00000000000000] [ 2.00000000000000 3.00000000000000] sage: numerical_approx(M) [0.000000000000000 0.500000000000000 0.666666666666667] [0.750000000000000 0.800000000000000 0.833333333333333] [0.857142857142857 0.875000000000000 0.888888888888889] [0.900000000000000 0.909090909090909 0.916666666666667] norm(p=2) Return the p-norm of this matrix, where p can be 1, 2, inf, or the Frobenius norm. INPUT: •self - a matrix whose entries are coercible into CDF •p - one of the following options: •1 - the largest column-sum norm •2 (default) - the Euclidean norm •Infinity - the largest row-sum norm •’frob’ - the Frobenius (sum of squares) norm OUTPUT: RDF number See Also: •sage.misc.functional.norm() EXAMPLES: sage: A = matrix(ZZ, [[1,2,4,3],[-1,0,3,-10]]) sage: A.norm(1) 13.0 sage: A.norm(Infinity) 14.0 sage: B = random_matrix(QQ, 20, 21) sage: B.norm(Infinity) == (B.transpose()).norm(1) True sage: Id = identity_matrix(12) sage: Id.norm(2) 1.0 sage: A = matrix(RR, 2, 2, [13,-4,-4,7]) sage: A.norm() 15.0 Norms of numerical matrices over high-precision reals are computed by this routine. Faster routines for double precision entries from RDF or CDF are provided by the Matrix_double_dense class. sage: A = matrix(CC, 2, 3, [3*I,4,1-I,1,2,0]) sage: A.norm(’frob’) 5.65685424949 sage: A.norm(2) 216 Chapter 7. Base class for matrices, part 2
Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 5.47068444321 sage: A.norm(1) 6.0 sage: A.norm(Infinity) 8.41421356237 sage: a = matrix([[],[],[],[]]) sage: a.norm() 0.0 sage: a.norm(Infinity) == a.norm(1) True nullity() Return the (left) nullity of this matrix, which is the dimension of the (left) kernel of this matrix acting from the right on row vectors. EXAMPLES: sage: M = Matrix(QQ,[[1,0,0,1],[0,1,1,0],[1,1,1,0]]) sage: M.nullity() 0 sage: M.left_nullity() 0 sage: A = M.transpose() sage: A.nullity() 1 sage: A.left_nullity() 1 sage: M = M.change_ring(ZZ) sage: M.nullity() 0 sage: A = M.transpose() sage: A.nullity() 1 numerical_approx(prec=None, digits=None) Return a numerical approximation of self as either a real or complex number with at least the requested number of bits or digits of precision. INPUT: •prec - an integer: the number of bits of precision •digits - an integer: digits of precision OUTPUT: A matrix coerced to a real or complex field with prec bits of precision. EXAMPLES: sage: d = matrix([[3, 0],[0,sqrt(2)]]) ; sage: b = matrix([[1, -1], [2, 2]]) ; e = b * d * b.inverse();e [ 1/2*sqrt(2) + 3/2 -1/4*sqrt(2) + 3/4] [ -sqrt(2) + 3 1/2*sqrt(2) + 3/2] sage: e.numerical_approx(53) [ 2.20710678118655 0.396446609406726] [ 1.58578643762690 2.20710678118655] 217
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Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

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