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Constructing a square matrix

  1. Aug 14, 2010 #1
    Hello,

    Given that we have some matrix M with unknown real elements a, b, c, d and we know its eigenvalues [tex]\lambda_{1}[/tex] and [tex]\lambda_{1}[/tex] (no eigenvectors though) and the value of the determinant is it possible to find the elements and hence the matrix M using this informaiton?

    Thanks
     
  2. jcsd
  3. Aug 14, 2010 #2
    No, there are infinite matrices with the same eigenvalues, but different eigenvectors. For example

    [tex]\left(\begin{array}{cc}1&2\\0&3\end{array}\right)\qquad\textrm{and}\qquad\left(\begin{array}{cc}3&4\\0&1\end{array}\right)[/tex]

    have the same eigenvalues, but are...different!

    By the way, the determinant is just the product of the eigenvalues, so it doesn't give further information.
     
    Last edited: Aug 14, 2010
  4. Aug 14, 2010 #3
    I think that the below is basically correct...

    Provided you have n eigenvalues for the nxn matrix, then if you know the eigenvalues then you know the matrix up to a similarity transformation.

    If you also know the eigenvectors then that gives you the similarity transformation and thus you know the matrix up to permutation of rows. (provided the eigenvectors are distinct)
     
  5. Aug 14, 2010 #4
    Actually in this case you know the matrix exactly, without any possibility of interchanging rows or columns, because if you change the order of the eigenvalues you have to change the order of the eigenvectors as well, and the result is always the same matrix.
     
  6. Aug 14, 2010 #5
    @Petr

    I knew (and have taught) that!
    That should teach me not to post whilst watching tv.
     
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