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# Complex Conjugate Matrix

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Tool to calculate the complex conjugate matrix. The complex conjugate of a matrix M is a matrix denoted $\overline{M}$ composed of the complex conjugate values of each element.
Summary

## Complexe Conjugate Matrix Calculator

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## Answers to Questions

### What is a complex matrix conjugate? (Definition)

The definition of a complex conjugate matrix is a matrix made of the conjugate elements of the matrix.

For the matrix $M=[a_{ij}]$, the conjugate matrix is noted with a bar $\overline{M}$ or with an asterisk $M^{*}$. For a complex value $z$, its conjugated value is written $\overline{z}$ or $z^{*}$. By generalizing, the formula for calculating the conjugate matrix is:

$$\overline{M} = [\overline{a_{ij}}] = [a_{ij}^{*}]$$

Remainder : the conjugate value of $a+ib$ is $a-ib$ (See the dCode page dedicated to complex conjugates)

### How to calculate the complex conjugate of a matrix?

The conjugate matrix is calculated for a matrix with complex elements by calculating the conjugate value of each element.

Example: $$M=\begin{bmatrix} 1 & 2-i \\ 3 & 4+2i \end{bmatrix} \Rightarrow \overline{M}= \begin{bmatrix} 1 & 2+i \\ 3 & 4-2i \end{bmatrix}$$

Use the character i to represent $i$ the imaginary unit for complex numbers.

### What are the properties of a conjugate matrix?

A double conjugated matrix (conjugated two times) is equal to the original matrix. $$\overline{\overline{M}}=M$$

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