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How write a matrix in terms of determinant and trace?

  1. May 19, 2014 #1
    Given the following: $$\\ \begin{bmatrix}
    A & 0\\
    0 & B\\
    \end{bmatrix}$$ the eigenvalues is exactaly A and B. So analogously, is possible to write a matrix with only two elements, T and D, such that the trace is T and the determinant is D?

    I tried something: $$\\ \text{tr} \left(
    \begin{bmatrix}
    \frac{1}{2}T & 0\\
    0 & \frac{1}{2}T\\
    \end{bmatrix}
    \right) = T $$
    $$\\ \text{det} \left(
    \begin{bmatrix}
    \sqrt{\frac{D}{2}} & \sqrt{-\frac{D}{2}} \\
    \sqrt{-\frac{D}{2}} & \sqrt{\frac{D}{2}} \\
    \end{bmatrix}
    \right) = D $$
    But I can't join the 2 formulas...
     
  2. jcsd
  3. May 19, 2014 #2

    Ben Niehoff

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    Science Advisor
    Gold Member

    There are infinitely many ways. You could write

    [tex]\begin{pmatrix}T & D \\ -1 & 0\end{pmatrix}[/tex]
    for example. Or you could try

    [tex]\begin{pmatrix}\frac{T}{2} & \frac{T}{2} + \sqrt D \\ \frac{T}{2} - \sqrt D & \frac{T}{2}\end{pmatrix}[/tex]
    Since ##T## and ##D## are the only invariants, these matrices ought to be similar. Maybe you can work out the similarity transformation.
     
  4. May 19, 2014 #3
    Yeah, yeah! The ideia is express a matrix in terms of the invariants, exist some general formula for this?
     
  5. May 20, 2014 #4

    Ben Niehoff

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    Gold Member

    No.

    4chars
     
  6. May 20, 2014 #5
    what?
     
  7. May 20, 2014 #6

    Matterwave

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    Science Advisor
    Gold Member

    He's saying "no". But his reply needs to be at least 4-characters long (to prevent spam), hence the "4chars" at the end.
     
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