6.48.5  Jordan normal form: jordan

The jordan command finds the Jordan form of a matrix.

• jordan takes one mandatory and one optional argument:
  • A, a square matrix.
  • Optionally, var, a variable name.
• jordan(A) (in all modes but Maple) returns a sequence [P,J] of two matrices, where the columns of P are the eigenvectors of A, J is the Jordan form of A, and

  J=P−1AP

  
  

  jordan(A), in Maple mode, only returns the matrix J.

• jordan(A ⟨var⟩) returns the matrix J, as above, and assigns the matrix P to the variable var.

Examples.

• Input (not in Maple mode):
  jordan([[4,1,1],[1,4,1],[1,1,4]])
  Output:

  ⎡ ⎢ ⎢ ⎣

  1 2 −1 1 0 2 1 −2 −1

  ⎤ ⎥ ⎥ ⎦

  ,

  ⎡ ⎢ ⎢ ⎣

  6 0 0 0 3 0 0 0 3

  ⎤ ⎥ ⎥ ⎦

• Input:
  jordan([[4,1,1],[1,4,1],[1,1,4]],P)
  Output:

  ⎡ ⎢ ⎢ ⎣

  6 0 0 0 3 0 0 0 3

  ⎤ ⎥ ⎥ ⎦

  then input:
  P
  Output:

  ⎡ ⎢ ⎢ ⎣

  1 2 −1 1 0 2 1 −2 −1

  ⎤ ⎥ ⎥ ⎦

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

Example:
Input:

Output: