# HG changeset patch # User Sebastian Pancratz # Date 1263953776 28800 # Node ID f2d8f1a29da3695146111e36ce58f405e5d93c1a # Parent a9ba19c1ec8e3c8173fc3fdc1c5445120af6600d Adding doctests for Jordan normal form computations, see 6942. diff -r a9ba19c1ec8e -r f2d8f1a29da3 sage/matrix/matrix2.pyx --- a/sage/matrix/matrix2.pyx Tue Jan 19 04:20:49 2010 -0800 +++ b/sage/matrix/matrix2.pyx Tue Jan 19 18:16:16 2010 -0800 @@ -5447,6 +5447,104 @@ sage: M=Matrix(1,1,[1]) sage: M.jordan_form(transformation=True) ([1], [1]) + + We now go through three `10 \times 10` matrices to exhibit cases where + there are multiple blocks of the same size:: + + sage: A = matrix(QQ, [[15, 37/3, -16, -104/3, -29, -7/3, 0, 2/3, -29/3, -1/3], [2, 9, -1, -6, -6, 0, 0, 0, -2, 0], [24, 74/3, -41, -208/3, -58, -23/3, 0, 4/3, -58/3, -2/3], [-6, -19, 3, 21, 19, 0, 0, 0, 6, 0], [2, 6, 3, -6, -3, 1, 0, 0, -2, 0], [-96, -296/3, 176, 832/3, 232, 101/3, 0, -16/3, 232/3, 8/3], [-4, -2/3, 21, 16/3, 4, 14/3, 3, -1/3, 4/3, -25/3], [20, 26/3, -66, -199/3, -42, -41/3, 0, 13/3, -55/3, -2/3], [18, 57, -9, -54, -57, 0, 0, 0, -15, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 3]]); A + [ 15 37/3 -16 -104/3 -29 -7/3 0 2/3 -29/3 -1/3] + [ 2 9 -1 -6 -6 0 0 0 -2 0] + [ 24 74/3 -41 -208/3 -58 -23/3 0 4/3 -58/3 -2/3] + [ -6 -19 3 21 19 0 0 0 6 0] + [ 2 6 3 -6 -3 1 0 0 -2 0] + [ -96 -296/3 176 832/3 232 101/3 0 -16/3 232/3 8/3] + [ -4 -2/3 21 16/3 4 14/3 3 -1/3 4/3 -25/3] + [ 20 26/3 -66 -199/3 -42 -41/3 0 13/3 -55/3 -2/3] + [ 18 57 -9 -54 -57 0 0 0 -15 0] + [ 0 0 0 0 0 0 0 0 0 3] + sage: J, T = A.jordan_form(transformation=True); J + [3 1 0|0 0 0|0 0 0|0] + [0 3 1|0 0 0|0 0 0|0] + [0 0 3|0 0 0|0 0 0|0] + [-----+-----+-----+-] + [0 0 0|3 1 0|0 0 0|0] + [0 0 0|0 3 1|0 0 0|0] + [0 0 0|0 0 3|0 0 0|0] + [-----+-----+-----+-] + [0 0 0|0 0 0|3 1 0|0] + [0 0 0|0 0 0|0 3 1|0] + [0 0 0|0 0 0|0 0 3|0] + [-----+-----+-----+-] + [0 0 0|0 0 0|0 0 0|3] + sage: T * J * T**(-1) == A + True + sage: T.rank() + 10 + + :: + + sage: A = matrix(QQ, [[15, 37/3, -16, -14/3, -29, -7/3, 0, 2/3, 1/3, 44/3], [2, 9, -1, 0, -6, 0, 0, 0, 0, 3], [24, 74/3, -41, -28/3, -58, -23/3, 0, 4/3, 2/3, 88/3], [-6, -19, 3, 3, 19, 0, 0, 0, 0, -9], [2, 6, 3, 0, -3, 1, 0, 0, 0, 3], [-96, -296/3, 176, 112/3, 232, 101/3, 0, -16/3, -8/3, -352/3], [-4, -2/3, 21, 16/3, 4, 14/3, 3, -1/3, 4/3, -25/3], [20, 26/3, -66, -28/3, -42, -41/3, 0, 13/3, 2/3, 82/3], [18, 57, -9, 0, -57, 0, 0, 0, 3, 28], [0, 0, 0, 0, 0, 0, 0, 0, 0, 3]]); A + [ 15 37/3 -16 -14/3 -29 -7/3 0 2/3 1/3 44/3] + [ 2 9 -1 0 -6 0 0 0 0 3] + [ 24 74/3 -41 -28/3 -58 -23/3 0 4/3 2/3 88/3] + [ -6 -19 3 3 19 0 0 0 0 -9] + [ 2 6 3 0 -3 1 0 0 0 3] + [ -96 -296/3 176 112/3 232 101/3 0 -16/3 -8/3 -352/3] + [ -4 -2/3 21 16/3 4 14/3 3 -1/3 4/3 -25/3] + [ 20 26/3 -66 -28/3 -42 -41/3 0 13/3 2/3 82/3] + [ 18 57 -9 0 -57 0 0 0 3 28] + [ 0 0 0 0 0 0 0 0 0 3] + sage: J, T = A.jordan_form(transformation=True); J + [3 1 0|0 0 0|0 0|0 0] + [0 3 1|0 0 0|0 0|0 0] + [0 0 3|0 0 0|0 0|0 0] + [-----+-----+---+---] + [0 0 0|3 1 0|0 0|0 0] + [0 0 0|0 3 1|0 0|0 0] + [0 0 0|0 0 3|0 0|0 0] + [-----+-----+---+---] + [0 0 0|0 0 0|3 1|0 0] + [0 0 0|0 0 0|0 3|0 0] + [-----+-----+---+---] + [0 0 0|0 0 0|0 0|3 1] + [0 0 0|0 0 0|0 0|0 3] + sage: T * J * T**(-1) == A + True + sage: T.rank() + 10 + + :: + + sage: A = matrix(QQ, [[15, 37/3, -16, -104/3, -29, -7/3, 35, 2/3, -29/3, -1/3], [2, 9, -1, -6, -6, 0, 7, 0, -2, 0], [24, 74/3, -29, -208/3, -58, -14/3, 70, 4/3, -58/3, -2/3], [-6, -19, 3, 21, 19, 0, -21, 0, 6, 0], [2, 6, -1, -6, -3, 0, 7, 0, -2, 0], [-96, -296/3, 128, 832/3, 232, 65/3, -279, -16/3, 232/3, 8/3], [0, 0, 0, 0, 0, 0, 3, 0, 0, 0], [20, 26/3, -30, -199/3, -42, -14/3, 70, 13/3, -55/3, -2/3], [18, 57, -9, -54, -57, 0, 63, 0, -15, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 3]]); A + [ 15 37/3 -16 -104/3 -29 -7/3 35 2/3 -29/3 -1/3] + [ 2 9 -1 -6 -6 0 7 0 -2 0] + [ 24 74/3 -29 -208/3 -58 -14/3 70 4/3 -58/3 -2/3] + [ -6 -19 3 21 19 0 -21 0 6 0] + [ 2 6 -1 -6 -3 0 7 0 -2 0] + [ -96 -296/3 128 832/3 232 65/3 -279 -16/3 232/3 8/3] + [ 0 0 0 0 0 0 3 0 0 0] + [ 20 26/3 -30 -199/3 -42 -14/3 70 13/3 -55/3 -2/3] + [ 18 57 -9 -54 -57 0 63 0 -15 0] + [ 0 0 0 0 0 0 0 0 0 3] + sage: J, T = A.jordan_form(transformation=True); J + [3 1 0|0 0|0 0|0 0|0] + [0 3 1|0 0|0 0|0 0|0] + [0 0 3|0 0|0 0|0 0|0] + [-----+---+---+---+-] + [0 0 0|3 1|0 0|0 0|0] + [0 0 0|0 3|0 0|0 0|0] + [-----+---+---+---+-] + [0 0 0|0 0|3 1|0 0|0] + [0 0 0|0 0|0 3|0 0|0] + [-----+---+---+---+-] + [0 0 0|0 0|0 0|3 1|0] + [0 0 0|0 0|0 0|0 3|0] + [-----+---+---+---+-] + [0 0 0|0 0|0 0|0 0|3] + sage: T * J * T**(-1) == A + True + sage: T.rank() + 10 """ from sage.matrix.constructor import block_diagonal_matrix, jordan_block, diagonal_matrix from sage.combinat.partition import Partition @@ -5553,38 +5651,6 @@ chain.reverse() Y.extend(chain) jordan_chains[eval].append(chain) - """ - if l == max(block_sizes): - # Let v be any vector in `\ker B^l` not in the kernel - # of `\ker B^{l-1}`, and start a chain from there. - v = _jordan_form_vector_in_difference(Vlarge, Vsmall) - chain = [v] - for i in range(l-1): - chain.append(B*chain[-1]) - chain.reverse() - Y.extend(chain) - jordan_chains[eval].append(chain) - - for i in range(1,count): - # Find v in `\ker B^l` but not in `\ker B^{l-1}` - # and also not in span(Y). - v = _jordan_form_vector_in_difference(Vlarge, Vsmall+Y) - chain = [v] - for i in range(l-1): - chain.append(B*chain[-1]) - chain.reverse() - Y.extend(chain) - jordan_chains[eval].append(chain) - else: - for i in range(count): - v = _jordan_form_vector_in_difference(Vlarge, Vsmall+Y) - chain = [v] - for i in range(l-1): - chain.append(B*chain[-1]) - chain.reverse() - Y.extend(chain) - jordan_chains[eval].append(chain) - """ # Now ``jordan_chains`` has all the columns of the transformation # matrix; we just need to put them in the right order.