|Jul1-05, 04:54 AM||#1|
Matrix Minimal Polynomial
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?
|Jul1-05, 06:04 AM||#2|
In other words the minimal polynomial has enough "stuff" to kill every vector, but does not have any extra "stuff". If The feild you are working in is algebraically closed (every polynomial has a root) as is the case with C the feild 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 feild factors may not exist in which case you keep the irreducible factors.
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