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NxN-complex matrix, identified 2Nx2N-real matrix, determinant

  1. Jul 5, 2011 #1
    If [itex]A\in\mathbb{C}^{N\times N}[/itex] is some complex matrix, is there anything we could say about the determinant of the matrix

    [tex]
    \left(\begin{array}{cc}
    \textrm{Re}(A) & -\textrm{Im}(A) \\
    \textrm{Im}(A) & \textrm{Re}(A) \\
    \end{array}\right)\quad\in\mathbb{R}^{2N\times 2N}
    [/tex]

    where [itex]\textrm{Re}(A)\in\mathbb{R}^{N\times N}[/itex] and [itex]\textrm{Im}(A)\in\mathbb{R}^{N\times N}[/itex] have been defined by element wise real and imaginary parts?

    For example, could it be that the determinant of the [itex]2N\times 2N[/itex]-matrix could be expressed as function of the determinants of the real [itex]N\times N[/itex]-matrices?

    Or is the asked determinant related to the complex determinant [itex]\det(A)\in\mathbb{C}[/itex]?

    I'm interested in this, because if I want to identify N-dimensional complex space with 2N-dimensional real space, then the complex linear transformation is naturally identified with the above matrix. The equation

    [tex]
    z^2 = Az^1,\quad\quad z^1,z^2\in\mathbb{C}^N
    [/tex]

    is equivalent with

    [tex]
    \left(\begin{array}{c}
    \textrm{Re}(z^2) \\ \textrm{Im}(z^2) \\
    \end{array}\right)
    = \left(\begin{array}{cc}
    \textrm{Re}(A) & -\textrm{Im}(A) \\
    \textrm{Im}(A) & \textrm{Re}(A) \\
    \end{array}\right)
    \left(\begin{array}{c}
    \textrm{Re}(z^1) \\ \textrm{Im}(z^1) \\
    \end{array}\right)
    [/tex]
     
  2. jcsd
  3. Jul 5, 2011 #2
    a conjecture

    I computed by brute force the following formula:

    [tex]
    \textrm{det} \left(\begin{array}{cccc}
    R_{11} & R_{12} & -I_{11} & -I_{12} \\
    R_{21} & R_{22} & -I_{21} & -I_{22} \\
    I_{11} & I_{12} & R_{11} & R_{12} \\
    I_{21} & I_{22} & R_{21} & R_{22} \\
    \end{array}\right)
    [/tex]
    [tex]
    = \textrm{det}(R)^2 + \textrm{det}(I)^2 + (R_{11}I_{22} - R_{21}I_{12})^2 + (R_{22}I_{11} - R_{12}I_{21})^2
    + 2(R_{11}I_{21} - R_{21}I_{11})(R_{22}I_{12} - R_{12}I_{22})
    [/tex]
    [tex]
    = \textrm{det}(R - iI) \textrm{det}(R + iI)
    [/tex]

    So it seems that

    [tex]
    \textrm{det}\left(\begin{array}{cc}
    \textrm{Re}(A) & -\textrm{Im}(A) \\
    \textrm{Im}(A) & \textrm{Re}(A) \\
    \end{array}\right) = |\textrm{det}(A)|^2
    [/tex]

    could be true for all [itex]A\in\mathbb{C}^{N\times N}[/itex].
     
  4. Dec 22, 2012 #3
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