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

[jostpuur]

If $A\in\mathbb{C}^{N\times N}$ is some complex matrix, is there anything we could say about the determinant of the matrix

$$\left(\begin{array}{cc}<br /> \textrm{Re}(A) & -\textrm{Im}(A) \\<br /> \textrm{Im}(A) & \textrm{Re}(A) \\<br /> \end{array}\right)\quad\in\mathbb{R}^{2N\times 2N}$$

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

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

Or is the asked determinant related to the complex determinant $\det(A)\in\mathbb{C}$?

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

$$z^2 = Az^1,\quad\quad z^1,z^2\in\mathbb{C}^N$$

is equivalent with

$$\left(\begin{array}{c}<br /> \textrm{Re}(z^2) \\ \textrm{Im}(z^2) \\<br /> \end{array}\right)<br /> = \left(\begin{array}{cc}<br /> \textrm{Re}(A) & -\textrm{Im}(A) \\<br /> \textrm{Im}(A) & \textrm{Re}(A) \\<br /> \end{array}\right)<br /> \left(\begin{array}{c}<br /> \textrm{Re}(z^1) \\ \textrm{Im}(z^1) \\<br /> \end{array}\right)$$

 

[jostpuur]

a conjecture

I computed by brute force the following formula:

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

So it seems that

$$\textrm{det}\left(\begin{array}{cc}<br /> \textrm{Re}(A) & -\textrm{Im}(A) \\<br /> \textrm{Im}(A) & \textrm{Re}(A) \\<br /> \end{array}\right) = |\textrm{det}(A)|^2$$

could be true for all $A\in\mathbb{C}^{N\times N}$.

 

[sfn17]

Yes, this is right.

There is some more to read in a math forum:

http://math.stackexchange.com/quest...plex-matrix-as-an-2n-times-2n-real-determinan