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 The subdivide option will add a natural subdivision between self and right. For more details about how subdivisions are managed when augmenting, see sage.matrix.matrix1.Matrix.augment(). sage: A = matrix(QQ, 3, 5, range(15), sparse=True) sage: B = matrix(QQ, 3, 3, range(9), sparse=True) sage: A.augment(B, subdivide=True) [ 0 1 2 3 4| 0 1 2] [ 5 6 7 8 9| 3 4 5] [10 11 12 13 14| 6 7 8] TESTS: Verify that Trac #12689 is fixed: sage: A = identity_matrix(QQ, 2, sparse=True) sage: B = identity_matrix(ZZ, 2, sparse=True) sage: A.augment(B) [1 0 1 0] [0 1 0 1] change_ring(ring) Return the matrix obtained by coercing the entries of this matrix into the given ring. Always returns a copy (unless self is immutable, in which case returns self). EXAMPLES: sage: A = matrix(QQ[’x,y’], 2, [0,-1,2*x,-2], sparse=True); A [ 0 -1] [2*x -2] sage: A.change_ring(QQ[’x,y,z’]) [ 0 -1] [2*x -2] Subdivisions are preserved when changing rings: sage: A.subdivide([2],[]); A [ 0 -1] [2*x -2] [-------] sage: A.change_ring(RR[’x,y’]) [ 0 -1.00000000000000] [2.00000000000000*x -2.00000000000000] [-------------------------------------] charpoly(var=’x’, **kwds) Return the characteristic polynomial of this matrix. Note - the generic sparse charpoly implementation in Sage is to just compute the charpoly of the corresponding dense matrix, so this could use a lot of memory. In particular, for this matrix, the charpoly will be computed using a dense algorithm. EXAMPLES: sage: A = matrix(ZZ, 4, range(16), sparse=True) sage: A.charpoly() x^4 - 30*x^3 - 80*x^2 sage: A.charpoly(’y’) y^4 - 30*y^3 - 80*y^2 sage: A.charpoly() x^4 - 30*x^3 - 80*x^2 284 Chapter 11. Base class for sparse matrices
Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 density() Return the density of the matrix. By density we understand the ratio of the number of nonzero positions and the self.nrows() * self.ncols(), i.e. the number of possible nonzero positions. EXAMPLES: sage: a = matrix([[],[],[],[]], sparse=True); a.density() 0 sage: a = matrix(5000,5000,{(1,2): 1}); a.density() 1/25000000 determinant(**kwds) Return the determinant of this matrix. Note: the generic sparse determinant implementation in Sage is to just compute the determinant of the corresponding dense matrix, so this could use a lot of memory. In particular, for this matrix, the determinant will be computed using a dense algorithm. EXAMPLES: sage: A = matrix(ZZ, 4, range(16), sparse=True) sage: B = A + identity_matrix(ZZ, 4, sparse=True) sage: B.det() -49 matrix_from_rows_and_columns(rows, columns) Return the matrix constructed from self from the given rows and columns. EXAMPLES: sage: M = MatrixSpace(Integers(8),3,3, sparse=True) sage: A = M(range(9)); A [0 1 2] [3 4 5] [6 7 0] sage: A.matrix_from_rows_and_columns([1], [0,2]) [3 5] sage: A.matrix_from_rows_and_columns([1,2], [1,2]) [4 5] [7 0] Note that row and column indices can be reordered or repeated: sage: A.matrix_from_rows_and_columns([2,1], [2,1]) [0 7] [5 4] For example here we take from row 1 columns 2 then 0 twice, and do this 3 times. sage: A.matrix_from_rows_and_columns([1,1,1],[2,0,0]) [5 3 3] [5 3 3] [5 3 3] We can efficiently extract large submatrices: sage: A = random_matrix(ZZ, 100000, density=.00005, sparse=True) sage: B = A[50000:,:50000] # long time # long time (4s on sage.ma 285
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Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

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