Minimal polynomial

15.2.9 Minimal polynomial

The minimal polynomial of a square matrix A is the polynomial P having minimal degree such that P(A)=0. The pmin command finds the minimal polynomial of a matrix.

pmin takes one mandatory argument and one optional argument:
- A, a square matrix.
- Optionally, x, a variable name.
pmin(A ⟨,x⟩) returns the minimal polynomial A. It is written as the list of its coefficients if no variable name was provided or written as an expression with respect to x if there is a second argument.

Examples

pmin([[1,0],[0,1]])

⎡
⎣

1,−1

⎤
⎦

pmin([[1,0],[0,1]],x)

x−1

Hence the minimal polynomial of [

1	0
0	1

] is x−1.

pmin([[2,1,0],[0,2,0],[0,0,2]])

⎡
⎣

1,−4,4

⎤
⎦

pmin([[2,1,0],[0,2,0],[0,0,2]],x)

x²−4 x+4

Hence, the minimal polynomial of [

2	1	0
0	2	0
0	0	2

] is x²−4x+4.