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Question about determinant

  1. Jul 9, 2010 #1

    quasar987

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    Let X and Y be n x n real matrices such that the 2n x 2n matrix M:=

    X -Y
    Y X

    is invertible. What is the relationship between det(M) and det(X+iY)??
     
  2. jcsd
  3. Jul 9, 2010 #2
    With a congruence transformation
    [tex]
    \begin{pmatrix}
    I &-iI\\-iI &I
    \end{pmatrix}
    \begin{pmatrix}
    X &-Y\\
    Y &X
    \end{pmatrix}
    \begin{pmatrix}
    I &iI\\iI &I
    \end{pmatrix}
    = 2\begin{pmatrix}
    X-iY &0\\0 &X+iY
    \end{pmatrix}
    [/tex]
    Hence the determinant is equal to the product determinant.
     
  4. Jul 10, 2010 #3

    quasar987

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    Ok, thanks!
     
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