- #1

jostpuur

- 2,116

- 19

[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]