An $11\times 11$ nilpotent matrix of index $4$.

{{{id=1| entries = [[4, 7, -6, 5, -3, -19, 6, -31, 27, -7, 20], [2, -9, 4, 7, -11, -3, -30, 9, -11, -8, 27], [-32, -25, 86, -56, 82, 139, 87, 235, -203, 58, -212], [7, 6, -18, 11, -17, -28, -20, -51, 44, -12, 37], [69, 48, -184, 123, -182, -304, -191, -497, 428, -132, 483], [15, 1, -32, 30, -45, -69, -51, -88, 73, -38, 136], [-16, -10, 42, -29, 43, 70, 47, 113, -97, 31, -114], [20, 8, -49, 39, -56, -90, -60, -132, 112, -43, 163], [21, 6, -51, 42, -61, -98, -62, -135, 114, -49, 186], [-38, -20, 96, -70, 104, 168, 114, 260, -222, 78, -284], [6, 1, -14, 12, -18, -27, -21, -37, 31, -14, 52]] /// }}} {{{id=2| NN = matrix(QQ, entries) NN /// [ 4 7 -6 5 -3 -19 6 -31 27 -7 20] [ 2 -9 4 7 -11 -3 -30 9 -11 -8 27] [ -32 -25 86 -56 82 139 87 235 -203 58 -212] [ 7 6 -18 11 -17 -28 -20 -51 44 -12 37] [ 69 48 -184 123 -182 -304 -191 -497 428 -132 483] [ 15 1 -32 30 -45 -69 -51 -88 73 -38 136] [ -16 -10 42 -29 43 70 47 113 -97 31 -114] [ 20 8 -49 39 -56 -90 -60 -132 112 -43 163] [ 21 6 -51 42 -61 -98 -62 -135 114 -49 186] [ -38 -20 96 -70 104 168 114 260 -222 78 -284] [ 6 1 -14 12 -18 -27 -21 -37 31 -14 52] }}} {{{id=3| Sequence([NN^i for i in [2,3,4]], cr=True) /// [ [ 2 -4 0 2 -6 0 -16 2 -4 -4 6] [ 2 -1 4 0 -2 -5 -3 -1 -1 -8 4] [ -10 10 -12 1 8 -12 48 -13 23 9 24] [ 2 0 5 -2 2 2 -2 2 -4 -2 -11] [ 21 -26 25 0 -21 28 -112 36 -57 -19 -40] [ 5 -13 5 4 -13 5 -41 17 -22 -9 14] [ -5 6 -6 0 5 -6 26 -8 13 5 9] [ 7 -12 6 3 -13 6 -46 13 -20 -10 3] [ 7 -16 5 5 -17 8 -56 19 -26 -10 12] [ -12 18 -14 -2 16 -14 70 -24 36 14 10] [ 2 -4 2 1 -4 2 -14 5 -7 -3 2], [ 0 12 6 -8 16 0 32 -12 12 4 -38] [ 0 2 6 -3 6 0 12 3 -3 -1 -8] [ 0 -84 -30 52 -104 0 -208 96 -96 -32 262] [ 0 16 6 -10 20 0 40 -18 18 6 -50] [ 0 182 63 -112 224 0 448 -210 210 70 -567] [ 0 40 15 -25 50 0 100 -45 45 15 -125] [ 0 -42 -15 26 -52 0 -104 48 -48 -16 131] [ 0 54 21 -34 68 0 136 -60 60 20 -169] [ 0 58 21 -36 72 0 144 -66 66 22 -181] [ 0 -100 -36 62 -124 0 -248 114 -114 -38 312] [ 0 16 6 -10 20 0 40 -18 18 6 -50], [0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0] ] }}}

Theorem ENLT:  All eigenvalues are zero.

{{{id=5| NN.eigenvalues() /// [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] }}}

Theorem DNLT: Nilpotent matrices are rarely diagonalizable.

{{{id=6| NN.right_eigenmatrix() /// ( [0 0 0 0 0 0 0 0 0 0 0] [ 1 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0] [ 0 1 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0] [ 0 0 1 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0] [ 0 0 0 1 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0] [ 0 -11/40 -203/80 -153/80 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0] [ 0 11/40 -77/80 -207/80 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0] [ 0 -1/40 47/80 37/80 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0] [ 1 1/40 -47/80 -37/80 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0] [ 1 -7/40 -71/80 -141/80 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0] [ 0 -3/20 61/40 71/40 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0], [ 0 1/10 -7/20 -17/20 0 0 0 0 0 0 0] ) }}}

Theorem KPLT, Theorem KPNLT:  Kernels of powers form an ascending chain which terminates at the index.

{{{id=4| nsp = [(NN^i).right_kernel() for i in range(12)] for i in range(12): print nsp[i] print print /// WARNING: Output truncated! full_output.txt Vector space of degree 11 and dimension 0 over Rational Field Basis matrix: [] Vector space of degree 11 and dimension 4 over Rational Field Basis matrix: [ 1 0 0 0 0 0 0 1 1 0 0] [ 0 1 0 0 -11/40 11/40 -1/40 1/40 -7/40 -3/20 1/10] [ 0 0 1 0 -203/80 -77/80 47/80 -47/80 -71/80 61/40 -7/20] [ 0 0 0 1 -153/80 -207/80 37/80 -37/80 -141/80 71/40 -17/20] Vector space of degree 11 and dimension 7 over Rational Field Basis matrix: [ 1 0 0 0 0 0 0 1 1 0 0] [ 0 1 0 0 0 0 0 -32/21 -37/21 3/7 2/7] [ 0 0 1 0 0 0 0 -41/21 -31/21 8/7 3/7] [ 0 0 0 1 0 0 0 25/21 23/21 -3/7 -2/7] [ 0 0 0 0 1 0 0 -71/21 -67/21 6/7 4/7] [ 0 0 0 0 0 1 0 4/3 5/3 -1 0] [ 0 0 0 0 0 0 1 -212/21 -211/21 19/7 8/7] Vector space of degree 11 and dimension 9 over Rational Field Basis matrix: [ 1 0 0 0 0 0 0 0 0 0 0] [ 0 1 0 0 0 0 0 0 0 -2/7 2/7] [ 0 0 1 0 0 0 0 0 0 18/7 3/7] [ 0 0 0 1 0 0 0 0 0 -5/7 -2/7] [ 0 0 0 0 1 0 0 0 0 10/7 4/7] [ 0 0 0 0 0 1 0 0 0 0 0] [ 0 0 0 0 0 0 1 0 0 20/7 8/7] [ 0 0 0 0 0 0 0 1 0 3 0] [ 0 0 0 0 0 0 0 0 1 -3 0] Vector space of degree 11 and dimension 11 over Rational Field Basis matrix: [1 0 0 0 0 0 0 0 0 0 0] [0 1 0 0 0 0 0 0 0 0 0] [0 0 1 0 0 0 0 0 0 0 0] [0 0 0 1 0 0 0 0 0 0 0] [0 0 0 0 1 0 0 0 0 0 0] [0 0 0 0 0 1 0 0 0 0 0] [0 0 0 0 0 0 1 0 0 0 0] [0 0 0 0 0 0 0 1 0 0 0] [0 0 0 0 0 0 0 0 1 0 0] [0 0 0 0 0 0 0 0 0 1 0] [0 0 0 0 0 0 0 0 0 0 1] Vector space of degree 11 and dimension 11 over Rational Field Basis matrix: [1 0 0 0 0 0 0 0 0 0 0] [0 1 0 0 0 0 0 0 0 0 0] [0 0 1 0 0 0 0 0 0 0 0] [0 0 0 1 0 0 0 0 0 0 0] [0 0 0 0 1 0 0 0 0 0 0] ... Vector space of degree 11 and dimension 11 over Rational Field Basis matrix: [1 0 0 0 0 0 0 0 0 0 0] [0 1 0 0 0 0 0 0 0 0 0] [0 0 1 0 0 0 0 0 0 0 0] [0 0 0 1 0 0 0 0 0 0 0] [0 0 0 0 1 0 0 0 0 0 0] [0 0 0 0 0 1 0 0 0 0 0] [0 0 0 0 0 0 1 0 0 0 0] [0 0 0 0 0 0 0 1 0 0 0] [0 0 0 0 0 0 0 0 1 0 0] [0 0 0 0 0 0 0 0 0 1 0] [0 0 0 0 0 0 0 0 0 0 1] Vector space of degree 11 and dimension 11 over Rational Field Basis matrix: [1 0 0 0 0 0 0 0 0 0 0] [0 1 0 0 0 0 0 0 0 0 0] [0 0 1 0 0 0 0 0 0 0 0] [0 0 0 1 0 0 0 0 0 0 0] [0 0 0 0 1 0 0 0 0 0 0] [0 0 0 0 0 1 0 0 0 0 0] [0 0 0 0 0 0 1 0 0 0 0] [0 0 0 0 0 0 0 1 0 0 0] [0 0 0 0 0 0 0 0 1 0 0] [0 0 0 0 0 0 0 0 0 1 0] [0 0 0 0 0 0 0 0 0 0 1] Vector space of degree 11 and dimension 11 over Rational Field Basis matrix: [1 0 0 0 0 0 0 0 0 0 0] [0 1 0 0 0 0 0 0 0 0 0] [0 0 1 0 0 0 0 0 0 0 0] [0 0 0 1 0 0 0 0 0 0 0] [0 0 0 0 1 0 0 0 0 0 0] [0 0 0 0 0 1 0 0 0 0 0] [0 0 0 0 0 0 1 0 0 0 0] [0 0 0 0 0 0 0 1 0 0 0] [0 0 0 0 0 0 0 0 1 0 0] [0 0 0 0 0 0 0 0 0 1 0] [0 0 0 0 0 0 0 0 0 0 1] Vector space of degree 11 and dimension 11 over Rational Field Basis matrix: [1 0 0 0 0 0 0 0 0 0 0] [0 1 0 0 0 0 0 0 0 0 0] [0 0 1 0 0 0 0 0 0 0 0] [0 0 0 1 0 0 0 0 0 0 0] [0 0 0 0 1 0 0 0 0 0 0] [0 0 0 0 0 1 0 0 0 0 0] [0 0 0 0 0 0 1 0 0 0 0] [0 0 0 0 0 0 0 1 0 0 0] [0 0 0 0 0 0 0 0 1 0 0] [0 0 0 0 0 0 0 0 0 1 0] [0 0 0 0 0 0 0 0 0 0 1] }}}

Ascending chain?

{{{id=7| [nsp[i].is_subspace(nsp[i+1]) for i in range(11)] /// [True, True, True, True, True, True, True, True, True, True, True] }}}

A basis of $\QQ^{11}$.

{{{id=13| R, T = NN.jordan_form(transformation=True) /// }}}

Columns of B are the basis.

{{{id=8| B=T.columns() /// }}}

R is the matrix representation.

{{{id=15| R /// [0 1 0 0|0 0 0 0|0 0|0] [0 0 1 0|0 0 0 0|0 0|0] [0 0 0 1|0 0 0 0|0 0|0] [0 0 0 0|0 0 0 0|0 0|0] [-------+-------+---+-] [0 0 0 0|0 1 0 0|0 0|0] [0 0 0 0|0 0 1 0|0 0|0] [0 0 0 0|0 0 0 1|0 0|0] [0 0 0 0|0 0 0 0|0 0|0] [-------+-------+---+-] [0 0 0 0|0 0 0 0|0 1|0] [0 0 0 0|0 0 0 0|0 0|0] [-------+-------+---+-] [0 0 0 0|0 0 0 0|0 0|0] }}} {{{id=16| B1 = [B[0], B[4], B[8], B[10]] BB1 = (QQ^11).subspace(B1) BB1 == NN.right_kernel() /// True }}} {{{id=17| [NN*B[1] == B[0], NN*B[5] == B[4], NN*B[9] == B[8]] /// [True, True, True] }}} {{{id=18| B2 = [B[0], B[4], B[8], B[10], B[1], B[5], B[9]] BB2 = (QQ^11).subspace(B2) BB2 == (NN^2).right_kernel() /// True }}} {{{id=19| [NN*B[2] == B[1], NN*B[6] == B[5]] /// [True, True] }}} {{{id=21| B3 = [B[0], B[4], B[8], B[10], B[1], B[5], B[9], B[2], B[6]] BB3 = (QQ^11).subspace(B3) BB3 == (NN^3).right_kernel() /// True }}} {{{id=22| [NN*B[3] == B[2], NN*B[7] == B[6]] /// [True, True] }}} {{{id=23| B4 = [B[0], B[4], B[8], B[10], B[1], B[5], B[9], B[2], B[6], B[3], B[7]] BB4 = (QQ^11).subspace(B4) BB4 == (NN^4).right_kernel() /// True }}}

Dimensions of Kernels

{{{id=30| [nsp[i].dimension() for i in range(12)] /// [0, 4, 7, 9, 11, 11, 11, 11, 11, 11, 11, 11] }}} {{{id=24| dims = [0]+[(NN^i).right_kernel().dimension() for i in range(1,11)] dimdiffs = [dims[i+1]-dims[i] for i in range(10)] dimdiffs /// [4, 3, 2, 2, 0, 0, 0, 0, 0, 0] }}} {{{id=28| /// }}}