Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Matrix Minimal Polynomial

  1. Jul 1, 2005 #1
    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
     
  2. jcsd
  3. Jul 1, 2005 #2

    lurflurf

    User Avatar
    Homework Helper

    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.
     
  4. Jul 1, 2005 #3

    mathwonk

    User Avatar
    Science Advisor
    Homework Helper
    2015 Award

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

    http://www.math.uga.edu/~roy/
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Matrix Minimal Polynomial
  1. Minimal polynomial (Replies: 10)

  2. The minimal polynomial (Replies: 3)

Loading...