Jordan forms of matrices over the rationals

Set A to be a matrix with entries in the rational field.

> M := MatrixRing(RationalField(), 10);
> A := M ! [i * j : i, j in [1 .. 10]];
> A;
[  1   2   3   4   5   6   7   8   9  10]
[  2   4   6   8  10  12  14  16  18  20]
[  3   6   9  12  15  18  21  24  27  30]
[  4   8  12  16  20  24  28  32  36  40]
[  5  10  15  20  25  30  35  40  45  50]
[  6  12  18  24  30  36  42  48  54  60]
[  7  14  21  28  35  42  49  56  63  70]
[  8  16  24  32  40  48  56  64  72  80]
[  9  18  27  36  45  54  63  72  81  90]
[ 10  20  30  40  50  60  70  80  90 100]

Note the characteristic polynomial and minimal polynomial of A.

> c<x> := CharacteristicPolynomial(A);
> c;
x^10 - 385*x^9
> Factorization(c);
[
    <x, 9>,
    <x - 385, 1>
]
> MinimalPolynomial(A);
x^2 - 385*x

Find the Jordan form J, with the transformation matrix T, and the primary invariant factors F.

> J, T, F := JordanForm(A);
> J;
[385   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]
> T;
[     1      2      3      4      5      6      7      8      9     10]
[     1      2      3      4      5      6      7      8      9  -57/2]
[     1      2      3      4      5      6      7      8 -304/9     10]
[     1      2      3      4      5      6      7 -321/8      9     10]
[     1      2      3      4      5      6    -48      8      9     10]
[     1      2      3      4      5 -349/6      7      8      9     10]
[     1      2      3      4    -72      6      7      8      9     10]
[     1      2      3 -369/4      5      6      7      8      9     10]
[     1      2 -376/3      4      5      6      7      8      9     10]
[  -384      2      3      4      5      6      7      8      9     10]
> F;
[
    <x - 385, 1>,
    <x, 1>,
    <x, 1>,
    <x, 1>,
    <x, 1>,
    <x, 1>,
    <x, 1>,
    <x, 1>,
    <x, 1>,
    <x, 1>
]

Check that T conjugates A to J.

> J eq T * A * T^-1;  
true

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