 
 
 
5.55.4  Rational Jordan matrix : rat_jordan
rat_jordan takes as argument a square
matrix A of size n with exact coefficients.
rat_jordan returns :
- 
in Xcas, Mupad or TI mode
 a sequence of two matrices : a matrix P (the columns of P are
the eigenvectors if A is diagonalizable in the field of its coefficients)
and the rational Jordan matrix J of A, that is the most reduced
matrix in the field of the coefficients of A (or the complexified
field in complex mode), where
- in Maple mode
 the Jordan matrix J of A. We can also have the matrix P verifying
J=P−1AP in a variable 
by passing this variable as second argument, for example rat_jordan([[1,0,0],[1,2,-1],[0,0,1]],’P’)
 
Remarks
- 
the syntax Maple is also valid in the other modes, for example, in
Xcas mode input 
 rat_jordan([[4,1,1],[1,4,1],[1,1,4]],’P’)
 Output : [[1,-1,1/2],[1,0,-1],[1,1,1/2]]
 then P returns [[6,0,0],[0,3,0],[0,0,3]]
 
- the coefficients of P and J belongs to the same field as the 
coefficients of A.
 For example, in Xcas mode, input : rat_jordan([[1,0,1],[0,2,-1],[1,-1,1]])
 Output : [[1,1,2],[0,0,-1],[0,1,2]],[[0,0,-1],[1,0,-3],[0,1,4]] Input (put -pcar(...) because the argument of companion is a unit 
polynomial (see 5.55.11) companion(-pcar([[1,0,1],[0,2,-1],[1,-1,1]],x),x) Output : [[0,0,-1],[1,0,-3],[0,1,4]] Input : rat_jordan([[1,0,0],[0,1,1],[1,1,-1]]) Output : [[-1,0,0],[1,1,1],[0,0,1]],[[1,0,0],[0,0,2],[0,1,0]] Input : factor(pcar([[1,0,0],[0,1,1],[1,1,-1]],x)) Output : -(x-1)*(x^2-2)
 Input : companion((x^2-2),x)
 Output : [[0,2],[1,0]] 
- When A is symmetric and has eigenvalues with an multiple order,
Xcas returns orthogonal eigenvectors (not always of norm equal to 1)
i.e. tran(P)*P is a diagonal matrix where the diagonal is the square norm
of the eigenvectors, for example :
 rat_jordan([[4,1,1],[1,4,1],[1,1,4]])
 returns : [[1,-1,1/2],[1,0,-1],[1,1,1/2]],[[6,0,0],[0,3,0],[0,0,3]]
 
 
Input in Xcas, Mupad or TI mode :
rat_jordan([[1,0,0],[1,2,-1],[0,0,1]])
Output :
[[0,1,0],[1,0,1],[0,1,1]],[[2,0,0],[0,1,0],[0,0,1]]
Input in Xcas, Mupad or TI mode :
rat_jordan([[4,1,-2],[1,2,-1],[2,1,0]])
Output :
[[[1,2,1],[0,1,0],[1,2,0]],[[2,1,0],[0,2,1],[0,0,2]]]
In complex mode and in Xcas, Mupad or TI mode , input :
rat_jordan([[2,0,0],[0,2,-1],[2,1,2]])
Output :
[[1,0,0],[-2,-1,-1],[0,-i,i]],[[2,0,0],[0,2-i,0],[0,0,2+i]]
Input in Maple mode :
rat_jordan([[1,0,0],[1,2,-1],[0,0,1]],’P’)
Output :
[[2,0,0],[0,1,0],[0,0,1]]
then input : 
P
Output :
[[0,1,0],[1,0,1],[0,1,1]]]
 
 
