### 5.48.5 Jordan normal form : jordan

jordan takes as argument a square matrix A of size n.
jordan returns :

- in Xcas, Mupad or TI mode
  a sequence of two matrices : a matrix P whose columns are the eigenvectors and characteristic vectors of the matrix A and the Jordan matrix J of A verifying J=P−1AP,
- 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
  jordan([[1,0,0],[0,1,1],[1,1,-1]],’P’)

Remarks

- the Maple syntax is also valid in the other modes, for example, in Xcas mode input :
  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]]
- When A is symmetric and has eigenvalues with multiple orders, 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 :
  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 :

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

Output :

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

Input in Maple mode :

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

Output :

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

then input :

P

Output :

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

Input in Xcas, Mupad or TI mode :

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 :

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]]