MinimalPolynomial.pdfHow Can I Find the Minimal Polynomial for a Given Matrix A?

[Chen]

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

 

[lurflurf]

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 constucted from the charateristic polynomial as follows. Take a root, if its multiplicity in the charateristic 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-1*I)^0)<1 (x-2) will have order 1
if nullity((A-1*I)^2)=2 and nullity((A-1*I)^1)<2 (x-3) will have order 2
Then the minimum polynomial is ((x-1)^2)((x-2)^1)((x-3)^2)
In short the charateristic 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 algenraically complete field factors may not exist in which case you keep the irreducible factors.

 

[mathwonk]

a theoretical discussion of minimal polynomials, and much more, is in the 15 page book on the website

http://www.math.uga.edu/~roy/