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Complex Conjugate Matrix

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.

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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