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 •Jean-Guillaume Dumas and Gilles Villars. ‘Computing the Rank of Large Sparse Matrices over Finite Fields’. Proc. CASC‘2002, The Fifth International Workshop on Computer Algebra in Scientific Computing, Big Yalta, Crimea, Ukraine, 22-27 sept. 2002, Springer-Verlag, http://perso.enslyon.fr/gilles.villard/BIBLIOGRAPHIE/POSTSCRIPT/rankjgd.ps Note: For very sparse matrices Gaussian elimination is faster because it barly has anything to do. If the fill in needs to be considered, ‘Symbolic Reordering’ is usually much faster. swap_rows(r1, r2) transpose() Return the transpose of self. EXAMPLE: sage: A = matrix(GF(127),3,3,[0,1,0,2,0,0,3,0,0],sparse=True) sage: A [0 1 0] [2 0 0] [3 0 0] sage: A.transpose() [0 2 3] [1 0 0] [0 0 0] .T is a convenient shortcut for the transpose: sage: A.T [0 2 3] [1 0 0] [0 0 0] visualize_structure(filename=None, maxsize=512) Write a PNG image to ‘filename’ which visualizes self by putting black pixels in those positions which have nonzero entries. White pixels are put at positions with zero entries. If ‘maxsize’ is given, then the maximal dimension in either x or y direction is set to ‘maxsize’ depending on which is bigger. If the image is scaled, the darkness of the pixel reflects how many of the represented entries are nonzero. So if e.g. one image pixel actually represents a 2x2 submatrix, the dot is darker the more of the four values are nonzero. INPUT: •filename - either a path or None in which case a filename in the current directory is chosen automatically (default:None) •maxsize - maximal dimension in either x or y direction of the resulting image. If None or a maxsize larger than max(self.nrows(),self.ncols()) is given the image will have the same pixelsize as the matrix dimensions (default: 512) EXAMPLES: sage: M = Matrix(GF(7), [[0,0,0,1,0,0,0,0],[0,1,0,0,0,0,1,0]], sparse=True); M [0 0 0 1 0 0 0 0] [0 1 0 0 0 0 1 0] sage: M.visualize_structure() 302 Chapter 15. Sparse matrices over Z/nZ for n small
CHAPTER SIXTEEN SYMBOLIC MATRICES Symbolic matrices Matrices with symbolic entries. The underlying representation is a pointer to a Maxima object. EXAMPLES: sage: matrix(SR, 2, 2, range(4)) [0 1] [2 3] sage: matrix(SR, 2, 2, var(’t’)) [t 0] [0 t] Arithmetic: sage: -matrix(SR, 2, range(4)) [ 0 -1] [-2 -3] sage: m = matrix(SR, 2, [1..4]); sqrt(2)*m [ sqrt(2) 2*sqrt(2)] [3*sqrt(2) 4*sqrt(2)] sage: m = matrix(SR, 4, [1..4^2]) sage: m * m [ 90 100 110 120] [202 228 254 280] [314 356 398 440] [426 484 542 600] sage: m = matrix(SR, 3, [1, 2, 3]); m [1] [2] [3] sage: m.transpose() * m [14] Computing inverses: sage: M = matrix(SR, 2, var(’a,b,c,d’)) sage: ~M [1/a - b*c/(a^2*(b*c/a - d)) b/(a*(b*c/a - d))] [ c/(a*(b*c/a - d)) -1/(b*c/a - d)] sage: (~M*M).simplify_rational() [1 0] [0 1] sage: M = matrix(SR, 3, 3, range(9)) - var(’t’) 303
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Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

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