Results 1 to 5 of 5

Math Help - Characteristic polynomial

  1. #1
    Junior Member
    Joined
    Feb 2006
    Posts
    32

    Characteristic polynomial

    Hi,

    Can anyone tell me the easiest way to find the characteristic polynomial of a 3x3 matrix.

    I don't understand why sometimes if you can reduce it to the identity matrix -the characteristic polynomial will just be (X-1)^3 and sometimes it won't . . . (thereby forcing you to use a different method to find the characteristic polynomial)

    Many thanks.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Apr 2006
    Posts
    401
    Quote Originally Posted by jedoob View Post
    Hi,

    Can anyone tell me the easiest way to find the characteristic polynomial of a 3x3 matrix.

    I don't understand why sometimes if you can reduce it to the identity matrix -the characteristic polynomial will just be (X-1)^3 and sometimes it won't . . . (thereby forcing you to use a different method to find the characteristic polynomial)

    Many thanks.
    (X-1)^3 (eigenvectors of 1);

    In order to find eigenvalues, you will become well acquainted with:

    Ax = Lx (where L is lambda);

    Similarly, (A - LI)x = 0, where I is the identity matrix, and x is the corresponding eigenvector.

    The characteristic polynomial is simply the part before the equality (= 0). The characteristic equation is then the whole thing. What you're interested in finding is the determinant of (A - LI)x. Do you have a specific problem that you're stuck on?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Feb 2006
    Posts
    32
    Thanks. I understand this, but what I don't understand is as follows:

    I have two different 3x3 matrices infront of me, they both reduce to become the identity matrix . . .

    BUT for some reason-the characteristic polynomial for one of them is (X-1)^3as expected but I know that this isn't true for the other one.

    Please could you explain why this should be so and also how I could find: det(LI-A) for a 3x3 matrix A directly, instead of reducing it to rref and then finding the determinent.
    [NOTE: I would also need to find the minimum polynomial afterwards so pref I need a method which will give me the det factorised as much as possible.]
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Nov 2006
    Posts
    2
    Well, you do not need to reduce anything in order to compute the characteristic polynomial. This is the determinant det(M-k*Id) by definition, so whatever method you like to compute a 3 x 3 determinant will do.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    9,930
    Thanks
    333
    Awards
    1
    Quote Originally Posted by jedoob View Post
    Thanks. I understand this, but what I don't understand is as follows:

    I have two different 3x3 matrices infront of me, they both reduce to become the identity matrix . . .

    BUT for some reason-the characteristic polynomial for one of them is (X-1)^3as expected but I know that this isn't true for the other one.

    Please could you explain why this should be so and also how I could find: det(LI-A) for a 3x3 matrix A directly, instead of reducing it to rref and then finding the determinent.
    [NOTE: I would also need to find the minimum polynomial afterwards so pref I need a method which will give me the det factorised as much as possible.]
    I'm curious. Could you post these examples?

    -Dan
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Characteristic Polynomial
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: November 28th 2011, 06:01 AM
  2. Characteristic Polynomial
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: April 21st 2010, 02:40 PM
  3. characteristic polynomial
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: April 6th 2010, 04:01 AM
  4. Replies: 1
    Last Post: December 15th 2009, 07:26 AM
  5. characteristic polynomial
    Posted in the Discrete Math Forum
    Replies: 0
    Last Post: November 9th 2008, 11:09 AM

Search Tags


/mathhelpforum @mathhelpforum