Coverage for local/lib/python2.7/site-packages/sage/tests/french_book/linalg_doctest.py : 0%

## -*- encoding: utf-8 -*- 

""" 

This file (./linalg_doctest.sage) was *autogenerated* from ./linalg.tex, 

with sagetex.sty version 2011/05/27 v2.3.1. 

It contains the contents of all the sageexample environments from this file. 

You should be able to doctest this file with: 

sage -t ./linalg_doctest.sage 

It is always safe to delete this file; it is not used in typesetting your 

document. 

Sage example in ./linalg.tex, line 136:: 

sage: MS = MatrixSpace(ZZ,2,3); MS 

Full MatrixSpace of 2 by 3 dense matrices over Integer Ring 

sage: VS = VectorSpace(GF(3^2,'x'),3); VS 

Vector space of dimension 3 over Finite Field in x of size 3^2 

Sage example in ./linalg.tex, line 148:: 

sage: list(MatrixSpace(ZZ,2,3).basis()) 

[ 

[1 0 0] [0 1 0] [0 0 1] [0 0 0] [0 0 0] [0 0 0] 

[0 0 0], [0 0 0], [0 0 0], [1 0 0], [0 1 0], [0 0 1] 

] 

Sage example in ./linalg.tex, line 190:: 

sage: A = matrix(GF(11), 2, 2, [1,0,0,2]) 

sage: B = matrix(GF(11), 2, 2, [0,1,1,0]) 

sage: MG = MatrixGroup([A,B]) 

sage: MG.cardinality() 

200 

sage: identity_matrix(GF(11),2) in MG 

True 

Sage example in ./linalg.tex, line 226:: 

sage: MS = MatrixSpace(ZZ,2,3); A = MS([1,2,3,4,5,6]); A 

[1 2 3] 

[4 5 6] 

Sage example in ./linalg.tex, line 247:: 

sage: a = matrix(); a.parent() 

Full MatrixSpace of 0 by 0 dense matrices over Integer Ring 

Sage example in ./linalg.tex, line 263:: 

sage: a = matrix(GF(8,'x'),3,4); a.parent() 

Full MatrixSpace of 3 by 4 dense matrices over Finite Field 

in x of size 2^3 

Sage example in ./linalg.tex, line 274:: 

sage: g = graphs.PetersenGraph() 

sage: m = matrix(g); m; m.parent() 

[0 1 0 0 1 1 0 0 0 0] 

[1 0 1 0 0 0 1 0 0 0] 

[0 1 0 1 0 0 0 1 0 0] 

[0 0 1 0 1 0 0 0 1 0] 

[1 0 0 1 0 0 0 0 0 1] 

[1 0 0 0 0 0 0 1 1 0] 

[0 1 0 0 0 0 0 0 1 1] 

[0 0 1 0 0 1 0 0 0 1] 

[0 0 0 1 0 1 1 0 0 0] 

[0 0 0 0 1 0 1 1 0 0] 

Full MatrixSpace of 10 by 10 dense matrices over Integer Ring 

Sage example in ./linalg.tex, line 295:: 

sage: A = matrix([[1,2],[3,4]]) 

sage: block_matrix([[A,-A],[2*A, A^2]]) 

[ 1 2|-1 -2] 

[ 3 4|-3 -4] 

[-----+-----] 

[ 2 4| 7 10] 

[ 6 8|15 22] 

Sage example in ./linalg.tex, line 320:: 

sage: A = matrix([[1,2,3],[4,5,6]]) 

sage: block_matrix([1,A,0,0,-A,2], ncols=3) 

[ 1 0| 1 2 3| 0 0] 

[ 0 1| 4 5 6| 0 0] 

[-----+--------+-----] 

[ 0 0|-1 -2 -3| 2 0] 

[ 0 0|-4 -5 -6| 0 2] 

Sage example in ./linalg.tex, line 346:: 

sage: A = matrix([[1,2,3],[0,1,0]]) 

sage: block_diagonal_matrix(A, A.transpose()) 

[1 2 3|0 0] 

[0 1 0|0 0] 

[-----+---] 

[0 0 0|1 0] 

[0 0 0|2 1] 

[0 0 0|3 0] 

Sage example in ./linalg.tex, line 397:: 

sage: A = matrix(3,3,range(9)) 

sage: A[:,1] = vector([1,1,1]); A 

[0 1 2] 

[3 1 5] 

[6 1 8] 

Sage example in ./linalg.tex, line 414:: 

sage: A[::-1], A[:,::-1], A[::2,-1] 

( 

[6 1 8] [2 1 0] 

[3 1 5] [5 1 3] [2] 

[0 1 2], [8 1 6], [8] 

) 

Sage example in ./linalg.tex, line 446:: 

sage: A = matrix(ZZ,4,4,range(16)); A 

[ 0 1 2 3] 

[ 4 5 6 7] 

[ 8 9 10 11] 

[12 13 14 15] 

Sage example in ./linalg.tex, line 462:: 

sage: A.matrix_from_rows_and_columns([0,2,3],[1,2]) 

[ 1 2] 

[ 9 10] 

[13 14] 

Sage example in ./linalg.tex, line 505:: 

sage: MS = MatrixSpace(GF(3),2,3) 

sage: MS.base_extend(GF(9,'x')) 

Full MatrixSpace of 2 by 3 dense matrices over Finite Field 

in x of size 3^2 

sage: MS = MatrixSpace(ZZ,2,3) 

sage: MS.change_ring(GF(3)) 

Full MatrixSpace of 2 by 3 dense matrices over Finite Field of size 3 

Sage example in ./linalg.tex, line 896:: 

sage: a = matrix(GF(7),4,3,[6,2,2,5,4,4,6,4,5,5,1,3]); a 

[6 2 2] 

[5 4 4] 

[6 4 5] 

[5 1 3] 

Sage example in ./linalg.tex, line 910:: 

sage: u = copy(identity_matrix(GF(7),4)); u[1:,0] = -a[1:,0]/a[0,0] 

sage: u, u*a 

( 

[1 0 0 0] [6 2 2] 

[5 1 0 0] [0 0 0] 

[6 0 1 0] [0 2 3] 

[5 0 0 1], [0 4 6] 

) 

Sage example in ./linalg.tex, line 932:: 

sage: v = copy(identity_matrix(GF(7),4)); v.swap_rows(1,2) 

sage: b = v*u*a; v, b 

( 

[1 0 0 0] [6 2 2] 

[0 0 1 0] [0 2 3] 

[0 1 0 0] [0 0 0] 

[0 0 0 1], [0 4 6] 

) 

Sage example in ./linalg.tex, line 954:: 

sage: w = copy(identity_matrix(GF(7),4)) 

sage: w[2:,1] = -b[2:,1]/b[1,1]; w, w*b 

( 

[1 0 0 0] [6 2 2] 

[0 1 0 0] [0 2 3] 

[0 0 1 0] [0 0 0] 

[0 5 0 1], [0 0 0] 

) 

Sage example in ./linalg.tex, line 1024:: 

sage: A = matrix(GF(7),4,5,[4,4,0,2,4,5,1,6,5,4,1,1,0,1,0,5,1,6,6,2]) 

sage: A, A.echelon_form() 

( 

[4 4 0 2 4] [1 0 5 0 3] 

[5 1 6 5 4] [0 1 2 0 6] 

[1 1 0 1 0] [0 0 0 1 5] 

[5 1 6 6 2], [0 0 0 0 0] 

) 

Sage example in ./linalg.tex, line 1147:: 

sage: a = matrix(ZZ, 4, 6, [2,1,2,2,2,-1,1,2,-1,2,1,-1,2,1,-1,\ 

....: -1,2,2,2,1,1,-1,-1,-1]); a.echelon_form() 

[ 1 2 0 5 4 -1] 

[ 0 3 0 2 -6 -7] 

[ 0 0 1 3 3 0] 

[ 0 0 0 6 9 3] 

Sage example in ./linalg.tex, line 1163:: 

sage: a.base_extend(QQ).echelon_form() 

[ 1 0 0 0 5/2 11/6] 

[ 0 1 0 0 -3 -8/3] 

[ 0 0 1 0 -3/2 -3/2] 

[ 0 0 0 1 3/2 1/2] 

Sage example in ./linalg.tex, line 1189:: 

sage: A = matrix(ZZ,4,5,[4,4,0,2,4,5,1,6,5,4,1,1,0,1,0,5,1,6,6,2]) 

sage: H, U = A.echelon_form(transformation=True); H, U 

( 

[ 1 1 0 0 2] [ 0 1 1 -1] 

[ 0 4 -6 0 -4] [ 0 -1 5 0] 

[ 0 0 0 1 -2] [ 0 -1 0 1] 

[ 0 0 0 0 0], [ 1 -2 -4 2] 

) 

Sage example in ./linalg.tex, line 1250:: 

sage: A = matrix(ZZ, 4, 5,\ 

....: [-1,-1,-1,-2,-2,-2,1,1,-1,2,2,2,2,2,-1,2,2,2,2,2]) 

sage: S,U,V = A.smith_form(); S,U,V 

( 

[ 0 -2 -1 -5 0] 

[1 0 0 0 0] [ 1 0 0 0] [ 1 0 1 -1 -1] 

[0 1 0 0 0] [ 0 0 1 0] [ 0 0 0 0 1] 

[0 0 3 0 0] [-2 1 0 0] [-1 2 0 5 0] 

[0 0 0 6 0], [ 0 0 -2 -1], [ 0 -1 0 -2 0] 

) 

Sage example in ./linalg.tex, line 1284:: 

sage: A.elementary_divisors() 

[1, 1, 3, 6] 

sage: S == U*A*V 

True 

Sage example in ./linalg.tex, line 1329:: 

sage: B = matrix(GF(7),5,4,[4,5,1,5,4,1,1,1,0,6,0,6,2,5,1,6,4,4,0,2]) 

sage: B.transpose().echelon_form() 

[1 0 5 0 3] 

[0 1 2 0 6] 

[0 0 0 1 5] 

[0 0 0 0 0] 

Sage example in ./linalg.tex, line 1344:: 

sage: B.pivot_rows() 

(0, 1, 3) 

sage: B.transpose().pivots() == B.pivot_rows() 

True 

Sage example in ./linalg.tex, line 1381:: 

sage: R.<x> = PolynomialRing(GF(5),'x') 

sage: A = random_matrix(R,2,3); A # random 

[ 3*x^2 + x x^2 + 2*x 2*x^2 + 2] 

[ x^2 + x + 2 2*x^2 + 4*x + 3 x^2 + 4*x + 3] 

Sage example in ./linalg.tex, line 1393:: 

sage: b = random_matrix(R,2,1); b # random 

[ 4*x^2 + 1] 

[3*x^2 + 2*x] 

Sage example in ./linalg.tex, line 1404:: 

sage: A.solve_right(b) # random 

[(4*x^3 + 2*x + 4)/(3*x^3 + 2*x^2 + 2*x)] 

[ (3*x^2 + 4*x + 3)/(x^3 + 4*x^2 + 4*x)] 

[ 0] 

Sage example in ./linalg.tex, line 1418:: 

sage: A.solve_right(b) == A\b 

True 

Sage example in ./linalg.tex, line 1449:: 

sage: a = matrix(QQ,3,5,[2,2,-1,-2,-1,2,-1,1,2,-1/2,2,-2,-1,2,-1/2]) 

sage: a.image() 

Vector space of degree 5 and dimension 3 over Rational Field 

Basis matrix: 

[ 1 0 0 1/4 -11/32] 

[ 0 1 0 -1 -1/8] 

[ 0 0 1 1/2 1/16] 

sage: a.right_kernel() 

Vector space of degree 5 and dimension 2 over Rational Field 

Basis matrix: 

[ 1 0 0 -1/3 8/3] 

[ 0 1 -1/2 11/12 2/3] 

Sage example in ./linalg.tex, line 1472:: 

sage: a = matrix(ZZ,5,3,[1,1,122,-1,-2,1,-188,2,1,1,-10,1,-1,-1,-1]) 

sage: a.kernel() 

Free module of degree 5 and rank 2 over Integer Ring 

Echelon basis matrix: 

[ 1 979 -11 -279 811] 

[ 0 2079 -22 -569 1488] 

sage: b = a.base_extend(QQ) 

sage: b.kernel() 

Vector space of degree 5 and dimension 2 over Rational Field 

Basis matrix: 

[ 1 0 -121/189 -2090/189 6949/63] 

[ 0 1 -2/189 -569/2079 496/693] 

sage: b.integer_kernel() 

Free module of degree 5 and rank 2 over Integer Ring 

Echelon basis matrix: 

[ 1 979 -11 -279 811] 

[ 0 2079 -22 -569 1488] 

Sage example in ./linalg.tex, line 1684:: 

sage: A = matrix(GF(97), 4, 4,\ 

....: [86,1,6,68,34,24,8,35,15,36,68,42,27,1,78,26]) 

sage: e1 = identity_matrix(GF(97),4)[0] 

sage: U = matrix(A.transpose().maxspin(e1)).transpose() 

sage: F = U^-1*A*U; F 

[ 0 0 0 83] 

[ 1 0 0 77] 

[ 0 1 0 20] 

[ 0 0 1 10] 

Sage example in ./linalg.tex, line 1703:: 

sage: K.<x> = GF(97)[] 

sage: P = x^4-sum(F[i,3]*x^i for i in range(4)); P 

x^4 + 87*x^3 + 77*x^2 + 20*x + 14 

Sage example in ./linalg.tex, line 1709:: 

sage: P == A.charpoly() 

True 

Sage example in ./linalg.tex, line 1813:: 

sage: A = matrix(ZZ,8,[[6,0,-2,4,0,0,0,-2],[14,-1,0,6,0,-1,-1,1],\ 

....: [2,2,0,1,0,0,1,0],[-12,0,5,-8,0,0,0,4],\ 

....: [0,4,0,0,0,0,4,0],[0,0,0,0,1,0,0,0],\ 

....: [-14,2,0,-6,0,2,2,-1],[-4,0,2,-4,0,0,0,4]]) 

sage: A.frobenius() 

[0 0 0 4 0 0 0 0] 

[1 0 0 4 0 0 0 0] 

[0 1 0 1 0 0 0 0] 

[0 0 1 0 0 0 0 0] 

[0 0 0 0 0 0 4 0] 

[0 0 0 0 1 0 0 0] 

[0 0 0 0 0 1 1 0] 

[0 0 0 0 0 0 0 2] 

Sage example in ./linalg.tex, line 1845:: 

sage: A.frobenius(1) 

[x^4 - x^2 - 4*x - 4, x^3 - x^2 - 4, x - 2] 

Sage example in ./linalg.tex, line 1851:: 

sage: F,K = A.frobenius(2) 

sage: K 

[ 1 -15/56 17/224 15/56 -17/896 0 -15/112 17/64] 

[ 0 29/224 -13/224 -23/448 -17/896 -17/896 29/448 13/128] 

[ 0 -75/896 75/896 -47/896 0 -17/896 -23/448 11/128] 

[ 0 17/896 -29/896 15/896 0 0 0 0] 

[ 0 0 0 0 1 0 0 0] 

[ 0 0 0 0 0 1 0 0] 

[ 0 1 0 0 0 0 1 0] 

[ 0 -4/21 -4/21 -10/21 0 0 -2/21 1] 

Sage example in ./linalg.tex, line 1877:: 

sage: K^-1*F*K == A 

True 

Sage example in ./linalg.tex, line 1905:: 

sage: S.<x> = QQ[] 

sage: B = x*identity_matrix(8) - A 

sage: B.elementary_divisors() 

[1, 1, 1, 1, 1, x - 2, x^3 - x^2 - 4, x^4 - x^2 - 4*x - 4] 

Sage example in ./linalg.tex, line 1913:: 

sage: A.frobenius(1) 

[x^4 - x^2 - 4*x - 4, x^3 - x^2 - 4, x - 2] 

Sage example in ./linalg.tex, line 1981:: 

sage: A = matrix(GF(7),4,[5,5,4,3,0,3,3,4,0,1,5,4,6,0,6,3]) 

sage: A.eigenvalues() 

[4, 1, 2, 2] 

sage: A.eigenvectors_right() 

[(4, [ 

(1, 5, 5, 1) 

], 1), (1, [ 

(0, 1, 1, 4) 

], 1), (2, [ 

(1, 3, 0, 1), 

(0, 0, 1, 1) 

], 2)] 

sage: A.eigenspaces_right() 

[ 

(4, Vector space of degree 4 and dimension 1 over Finite Field 

of size 7 

User basis matrix: 

[1 5 5 1]), 

(1, Vector space of degree 4 and dimension 1 over Finite Field 

of size 7 

User basis matrix: 

[0 1 1 4]), 

(2, Vector space of degree 4 and dimension 2 over Finite Field 

of size 7 

User basis matrix: 

[1 3 0 1] 

[0 0 1 1]) 

] 

Sage example in ./linalg.tex, line 2019:: 

sage: A.eigenmatrix_right() 

( 

[4 0 0 0] [1 0 1 0] 

[0 1 0 0] [5 1 3 0] 

[0 0 2 0] [5 1 0 1] 

[0 0 0 2], [1 4 1 1] 

) 

Sage example in ./linalg.tex, line 2144:: 

sage: A = matrix(ZZ,4,[3,-1,0,-1,0,2,0,-1,1,-1,2,0,1,-1,-1,3]) 

sage: A.jordan_form() 

[3|0|0 0] 

[-+-+---] 

[0|3|0 0] 

[-+-+---] 

[0|0|2 1] 

[0|0|0 2] 

Sage example in ./linalg.tex, line 2163:: 

sage: J,U = A.jordan_form(transformation=True) 

sage: U^-1*A*U == J 

True 

""" 

# This file was *autogenerated* from the file linalg_doctest.sage. 

from sage.all_cmdline import * # import sage library