Results 1 to 4 of 4

Math Help - Eigenvalues and eigenvectors of a 3x3 matrix A? A unknown?

  1. #1
    Junior Member
    Joined
    Feb 2010
    Posts
    54

    Eigenvalues and eigenvectors of a 3x3 matrix A? A unknown?

    I need some help with the following problem please?

    Let A be a 3x3 matrix with eigenvalues -1,0,1 and corresponding eigenvectors
    l1l . l0l . l0l
    l0l ; l1l ; l1l respectively.
    l0l . l1l . l2l

    Find A.

    I know that I need to work backwards on this problem so I set up the characteristic equation with th known eigenvalues ending up with x^3-x=0 but now I'm stuck I don't know where to go from here and how to use the eigenvectors. I can set up A with all unknowns and subtract it from kI where k is the eigenvalue and I is the identity matrix then find the determinant, but there are too many unknowns and too few equations. I'm totally confused.
    Can someone please assist?

    Thanks in advance.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Quote Originally Posted by chocaholic View Post
    I need some help with the following problem please?

    Let A be a 3x3 matrix with eigenvalues -1,0,1 and corresponding eigenvectors
    l1l . l0l . l0l
    l0l ; l1l ; l1l respectively.
    l0l . l1l . l2l

    Find A.

    I know that I need to work backwards on this problem so I set up the characteristic equation with th known eigenvalues ending up with x^3-x=0 but now I'm stuck I don't know where to go from here and how to use the eigenvectors. I can set up A with all unknowns and subtract it from kI where k is the eigenvalue and I is the identity matrix then find the determinant, but there are too many unknowns and too few equations. I'm totally confused.
    Can someone please assist?

    Thanks in advance.

    Put P=\begin{pmatrix}1&0&0\\0&1&1\\0&1&2\end{pmatrix}\  Longrightarrow PAP^{-1}=\begin{pmatrix}\!\!-1&0&1\\\;0&0&0\\\;0&0&1\end{pmatrix} ...and now find A.

    Tonio
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Feb 2010
    Posts
    54
    Hi Tonio isn't that the formula for diagonalization? I don't understand how it relates here and how you got the answer for PAP^-1 ? Further insight would be appreciated.
    Thanks
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Quote Originally Posted by chocaholic View Post
    Hi Tonio isn't that the formula for diagonalization? I don't understand how it relates here and how you got the answer for PAP^-1 ? Further insight would be appreciated.
    Thanks

    Of course that's the formula for diagonalization! We know A is diagonalizable because we're given three LINEAR INDEPENDENT eigenvectors (how we know they're lin. indep. without directly checking this?), which are then a basis for the vector space, and thus we can apply that formula...

    Now, sometimes the formula is P^{-1}AP=D and other times it is PAP^{-1}=D , with D= the diagonal matrix with the eigenvalues on its main diagonal (it all depends on how we define a matrix of a trnasformation), but since we know what is  D we can now find A .

    Tonio
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: December 6th 2010, 01:00 PM
  2. eigenvalues/eigenvectors of a 3x3 matrix
    Posted in the Algebra Forum
    Replies: 4
    Last Post: May 19th 2010, 02:48 AM
  3. Finding eigenvalues and eigenvectors of a real matrix
    Posted in the Advanced Algebra Forum
    Replies: 7
    Last Post: August 18th 2009, 06:05 AM
  4. Eigenvectors and eigenvalues of a matrix
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: May 6th 2008, 08:29 AM
  5. Replies: 1
    Last Post: May 6th 2008, 07:24 AM

Search Tags


/mathhelpforum @mathhelpforum