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Thread: polynomial, eigenvalues, eigenvectors, diagonal matrix

  1. #1
    Jan 2009

    Talking polynomial, eigenvalues, eigenvectors, diagonal matrix

    For matrix A$\displaystyle =\begin{pmatrix}-4 & 1 & 1\\1 & 5 & -1\\0 & 1 & -3\end{pmatrix}$
    a) Find the characteristic polynomial of A
    b) Find the eigenvalues of A
    c) Find the eigenvectors of A
    d) If A is diagonalizable, find a matrix P such that $\displaystyle P^{-1}AP$ is diagonal

    I am stressing out because I really am not understanding any of this. Thank you in advance for any help!
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  2. #2
    Super Member Gamma's Avatar
    Dec 2008
    Iowa City, IA

    here you go

    1) to find the characteristic polynomial you just do the $\displaystyle det(A-xI_n)$ subtract x from all the diagonal entries and find the determinant. I got (x+4)(x-5)(x+3)

    2) The eigenvalues are the roots to the characteristic polynomial. $\displaystyle \lambda_1=-4, \lambda_2=5, \lambda_3=-3$

    3) you just gotta solve $\displaystyle (A- \lambda_iI_n)v_i=0$ for each of the eigenvalues. They are distinct, so you will get solutions to all of these, the $\displaystyle v_i$ are the corresponding eigenvectors for the eigenvalues.
    I got $\displaystyle v_1=<-10,1,-1>, v_2=<1,8,1>, v_3=<1,0,1>$

    4) Your P is going to just be the collection of these eigenvectors, so the matrix with ith column $\displaystyle v_i$.

    I don't know if you have talked about Jordan Canonical Form, but as soon as you see it has 3 distinct eigenvalues you can be sure it is diagonalizable, and this is your matrix that you use to get it into its diagonal form. It should equal the matrix with diagonal entries -4, 5, -3, the eigenvalues. And in this order. This is unique up to permutation of the diagonal elements. You should check all these calculations for understanding at each step as I did them kinda fast, but this should be a good start.
    Last edited by Gamma; Apr 20th 2009 at 08:21 PM. Reason: forgot the identity matrix
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