Thread Closed

Matrix Minimal Polynomial

 
Share Thread Thread Tools
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?

Thanks,
Chen
PhysOrg.com
PhysOrg
science news on PhysOrg.com

>> King Richard III found in 'untidy lozenge-shaped grave'
>> Google Drive sports new view and scan enhancements
>> Researcher admits mistakes in stem cell study
Jul1-05, 06:04 AM   #2
 
Recognitions:
Homework Helper Homework Help
Quote by 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
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 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.
Jul1-05, 12:40 PM   #3
 
Recognitions:
Homework Helper Homework Help
Science Advisor Science Advisor
a theoretical discussion of minimal polynomials, and much more, is in the 15 page book on the website

http://www.math.uga.edu/~roy/
Thread Closed
Thread Tools


Similar Threads for: Matrix Minimal Polynomial
Thread Forum Replies
How to convert polynomial to matrix??? Linear & Abstract Algebra 3
Minimal Polynomial A nxn Matrix Calculus & Beyond Homework 5
a question on minimal polynomial (LA) Calculus & Beyond Homework 5
Minimal polynomial Linear & Abstract Algebra 10
Minimal and characteristic polynomial Linear & Abstract Algebra 4