Rational Jordan matrix

15.2.4 Rational Jordan matrix

The rat_jordan command finds the rational Jordan form of a matrix.

rat_jordan takes one mandatory and one optional argument:
- A, a square matrix (preferably with exact coefficients).
- Optionally, var, a variable name.
rat_jordan(A) (in all modes but Maple) returns a sequence [P,J] of two matrices, where J is the rational Jordan matrix of A (the most reduced matrix in the field of the coefficients of A or the complexified field in complex mode) and
J=P⁻¹AP

The coefficients of P and J belongs to the same field as the coefficients of A. If A is diagonalizable in the field of its coefficients, then the columns of P are the eigenvectors of A.
rat_jordan(A) (in Maple mode) only returns the matrix J.
rat_jordan(A ⟨,var ⟩) returns the matrix J, as above, and assigns the matrix P to the variable var.

Examples

Input not in Maple mode:

rat_jordan([[1,0,0],[1,2,-1],[0,0,1]])

⎡
⎢
⎢
⎣

0	1	0
1	0	1
0	1	1

⎤
⎥
⎥
⎦

⎡
⎢
⎢
⎣

2	0	0
0	1	0
0	0	1

⎤
⎥
⎥
⎦

rat_jordan([[1,0,0],[1,2,-1],[0,0,1]],P)

⎡
⎢
⎢
⎣

2	0	0
0	1	0
0	0	1

⎤
⎥
⎥
⎦

⎡
⎢
⎢
⎣

0	1	0
1	0	1
0	1	1

⎤
⎥
⎥
⎦

rat_jordan([[1,0,1],[0,2,-1],[1,-1,1]])

⎡
⎢
⎢
⎣

1	1	2
0	0	−1
0	1	2

⎤
⎥
⎥
⎦

⎡
⎢
⎢
⎣

0	0	−1
1	0	−3
0	1	4

⎤
⎥
⎥
⎦

rat_jordan([[1,0,0],[0,1,1],[1,1,-1]])

⎡
⎢
⎢
⎣

−1	0	0
1	1	1
0	0	1

⎤
⎥
⎥
⎦

⎡
⎢
⎢
⎣

1	0	0
0	0	2
0	1	0

⎤
⎥
⎥
⎦

If A is symmetric and has eigenvalues with multiple orders, the matrix P returned by rat_jordan(A) will contain orthogonal eigenvectors (not always of norm equal to 1); that is, P^TP will be a diagonal matrix where the diagonal is the square norm of the eigenvectors.

rat_jordan([[4,1,1],[1,4,1],[1,1,4]])

⎡
⎢
⎢
⎣

1	2	−1
1	0	2
1	−2	−1

⎤
⎥
⎥
⎦

⎡
⎢
⎢
⎣

6	0	0
0	3	0
0	0	3

⎤
⎥
⎥
⎦