Jordan form of a matrix

15.2.5 Jordan form of a matrix

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]])

⎡
⎢
⎢
⎣

1	2	−1
1	0	2
1	−2	−1

⎤
⎥
⎥
⎦

⎡
⎢
⎢
⎣

6	0	0
0	3	0
0	0	3

⎤
⎥
⎥
⎦

jordan([[4,1,1],[1,4,1],[1,1,4]],P)

⎡
⎢
⎢
⎣

6	0	0
0	3	0
0	0	3

⎤
⎥
⎥
⎦

⎡
⎢
⎢
⎣

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); that is, P^TP will be a diagonal matrix where the diagonal is the square norm of the eigenvectors.

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

⎤
⎥
⎥
⎦