Sage Reference Manual: Matrices and Spaces of Matrices

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Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.2sage: p = t^3 + 3*t - 8sage: q = t^5 + t - 17sage: A = block_diagonal_matrix( companion_matrix(p),... companion_matrix(p^2),... companion_matrix(q),... companion_matrix(q) )sage: A.charpoly(var=t).factor()(t^3 + 3*t - 8)^3 * (t^5 + t - 17)^2sage: A.minpoly(var=t).factor()(t^3 + 3*t - 8)^2 * (t^5 + t - 17)TESTS:sage: companion_matrix([4, 5, 1], format=’junk’)Traceback (most recent call last):...ValueError: format must be ’right’, ’left’, ’top’ or ’bottom’, not junksage: companion_matrix(sin(x))Traceback (most recent call last):...TypeError: input must be a polynomial (not a symbolic expression, see docstring), or other iterasage: companion_matrix([2, 3, 896])Traceback (most recent call last):...ValueError: polynomial (or the polynomial implied by coefficients) must be monic, not a leadingsage: F. = GF(2^2)sage: companion_matrix([4/3, a+1, 1])Traceback (most recent call last):...TypeError: unable to find common ring for coefficients from polynomialsage: A = companion_matrix([1])sage: A.nrows(); A.ncols()00sage: A = companion_matrix([])Traceback (most recent call last):...ValueError: polynomial cannot be specified by an empty listAUTHOR:•Rob Beezer (2011-05-19)sage.matrix.constructor.diagonal_matrix(arg0=None, arg1=None, arg2=None,sparse=True)This function is available as diagonal_matrix(...) and matrix.diagonal(...).Return a square matrix with specified diagonal entries, and zeros elsewhere.FORMATS:1.diagonal_matrix(entries)2.diagonal_matrix(nrows, entries)3.diagonal_matrix(ring, entries)28 Chapter 2. Matrix Constructor
Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.24.diagonal_matrix(ring, nrows, entries)INPUT:•entries - the values to place along the diagonal of the returned matrix. This may be a flat list, a flattuple, a vector or free module element, or a one-dimensional NumPy array.•nrows - the size of the returned matrix, which will have an equal number of columns•ring - the ring containing the entries of the diagonal entries. This may not be specified in combinationwith a NumPy array.•sparse - default: True - whether or not the result has a sparse implementation.OUTPUT:A square matrix over the given ring with a size given by nrows. If the ring is not given it is inferred from thegiven entries. The values on the diagonal of the returned matrix come from entries. If the number of entriesis not enough to fill the whole diagonal, it is padded with zeros.EXAMPLES:We first demonstrate each of the input formats with various different ways to specify the entries.Format 1: a flat list of entries.sage: A = diagonal_matrix([2, 1.3, 5]); A[ 2.00000000000000 0.000000000000000 0.000000000000000][0.000000000000000 1.30000000000000 0.000000000000000][0.000000000000000 0.000000000000000 5.00000000000000]sage: A.parent()Full MatrixSpace of 3 by 3 sparse matrices over Real Field with 53 bits of precisionFormat 2: size specified, a tuple with initial entries. Note that a short list of entries is effectively padded withzeros.sage: A = diagonal_matrix(3, (4, 5)); A[4 0 0][0 5 0][0 0 0]sage: A.parent()Full MatrixSpace of 3 by 3 sparse matrices over Integer RingFormat 3: ring specified, a vector of entries.sage: A = diagonal_matrix(QQ, vector(ZZ, [1,2,3])); A[1 0 0][0 2 0][0 0 3]sage: A.parent()Full MatrixSpace of 3 by 3 sparse matrices over Rational FieldFormat 4: ring, size and list of entries.sage: A = diagonal_matrix(FiniteField(3), 3, [2, 16]); A[2 0 0][0 1 0][0 0 0]sage: A.parent()Full MatrixSpace of 3 by 3 sparse matrices over Finite Field of size 3NumPy arrays may be used as input.29
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Sage Reference Manual: Matrices and Spaces of Matrices

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