# MinPoly(M) = Minimal polynomial of square matrix or scalar M

MinPoly(M) = Minimal polynomial of square matrix or scalar M:

The minimal polynomial of a square matrix M is the monic polynomial P of smallest degree for which P(M) = 0. The characteristic polynomial CP will always have CP(M) = 0 and P will divide CP with no remainder.

See: http://www.physicsforums.com/archive/index.php/t-80841.html :

Given a matrix A how can I found its minimal polynomial? I know how to find its characteristic polynomial, but how do.I reduce it to minimal?
Thanks,
Chen

If A is a matrix and for every polynomial q such that q(A)=0 p|q for some monic polynomial p, then p is the minimal of A.
In other words the minimal polynomial has enough "stuff" to kill every vector, but does not have any extra "stuff". If the field you are working in is algebraically closed (every polynomial has a root) as is the case with C the field of complex numbers things are relatively simple.

The characteristic polynomial can be factored (at least in principle).
The characteristic and minimal polynomials have the same roots but the roots may have different multiplicities. The minimal polynomial can be constructed from the characteristic polynomial as follows. Take a root, if its multiplicity in the characteristic polynomial is n then its multiplicity in the minimal polynomial is the smallest k such that nullity((A−root*I)^k)=n. An example might help:

Say for some matrix A the characteristic polynomial is ((x−1)^4)((x−2)^3)((x−3)^2)
If nullity((A−1*I)^2)=4 and nullity((A−1*I)^1)<4 (x−1) will have order 2
If nullity((A−2*I)^1)=3 and nullity((A−2*I)^0)<3 (x−2) will have order 1
If nullity((A−3*I)^2)=2 and nullity((A−3*I)^1)<2 (x−3) will have order 2
Then the minimal polynomial is ((x−1)^2)((x−2)^1)((x−3)^2).

In short the characteristic polynomial with kill all vectors, the minimal polynomial also kills all vectors but it may lack some factors of the characteristic polynomial that are not need for killing vectors. If you are not working in an algebraically complete field factors may not exist in which case you keep the irreducible factors.

Note: The nullity of a matrix is the dimension of its null space. For a square n by n matrix, its nullity is n minus its rank.

If M is a scalar or a one by one matrix, MinPoly(M) = CharPoly(M).