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

• jostpuur
In summary, the determinant of the 2N\times 2N-matrix \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} can be expressed as the square of the determinant of the complex N\times N-matrix A\in\mathbb{C}^{N\times N}. This can be seen by comparing it to the equation z^2 = Az^1,\quad\quad z^1
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

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

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

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

a conjecture

I computed by brute force the following formula:

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

So it seems that

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

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

## 1. What is an NxN-complex matrix?

An NxN-complex matrix is a mathematical structure that consists of a rectangular array of complex numbers, where the number of rows is equal to the number of columns (NxN). It is represented as a square matrix with complex numbers as its elements.

## 2. What is the difference between an NxN-complex matrix and a 2Nx2N-real matrix?

An NxN-complex matrix is a square matrix with complex numbers as its elements, whereas a 2Nx2N-real matrix is a rectangular matrix with real numbers as its elements. The number of rows and columns in a 2Nx2N-real matrix is twice the number of rows and columns in an NxN-complex matrix.

## 3. How is the determinant of an NxN-complex matrix calculated?

The determinant of an NxN-complex matrix is calculated using the same method as a determinant of any square matrix. It involves finding the product of the elements in the main diagonal and subtracting the product of the elements in the other diagonal. The result is a complex number.

## 4. Can the determinant of a 2Nx2N-real matrix be a complex number?

Yes, the determinant of a 2Nx2N-real matrix can be a complex number. This is because the determinant of a matrix is calculated using complex numbers, regardless of whether the matrix itself is composed of real or complex numbers.

## 5. What is the significance of the determinant of an NxN-complex matrix?

The determinant of an NxN-complex matrix is a useful tool in linear algebra as it provides information about the matrix, such as its invertibility and its relationship to other matrices. It is also used to solve systems of linear equations and to calculate eigenvalues and eigenvectors.

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