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.
Input:
pmin([[1,0],[0,1]])
Output:
⎡
⎣
1,−1
⎤
⎦
Input:
pmin([[1,0],[0,1]],x)
Output:
x−1
Hence the minimal polynomial of [[1,0],[0,1]] is x−1.
Input:
pmin([[2,1,0],[0,2,0],[0,0,2]])
Output:
⎡
⎣
1,−4,4
⎤
⎦
Input:
pmin([[2,1,0],[0,2,0],[0,0,2]],x)
Output:
x2−4 x+4
Hence, the minimal polynomial of [[2,1,0],[0,2,0],[0,0,2]] is x2−4x+4.