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Conjugate Transpose Matrix

Tool to calculate the conjugate transpose matrix (or Hermitian transpose matrix), the transpose of the conjugate matrix of a complex matrix M.

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Conjugate Transpose Matrix -

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# Conjugate Transpose Matrix

## Hermitian Matrix Checker

Calculate the Conjugate Transpose matrix if it is equal to the initial matrix then the matrix is Hermitian.

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

The conjugate transpose matrix is the name given to the transpose of the conjugate of a complex matrix (or of the conjugate of the transposed matrix), it is denoted $M^*$ (asterisk) or, more rare notation, with a dagger † $M^\dagger$.

### How to calculate the conjugate transpose of a matrix? (Formula)

Taking $M=[a_{ij}]$ a matrix with complex elements, the conjugate transpose matrix is computed with the formula $$M^* = \overline{M}^T = \overline{M^T} = [\overline{a_{ij}}]^T$$

Example: The conjugate transpose 2x2 matrix $M^*$ of the matrix $M$ is calculated: $$M=\begin{bmatrix} 2 & 1-i & 0 \\ 1 & 2+i & -i \end{bmatrix} \Rightarrow M^*= \begin{bmatrix} 2 & 1 \\ 1+i & 2-i \\ 0 & i \end{bmatrix}$$

On dCode, use the character i to represent the imaginary unit $i$ of complex numbers.

### What is the hermitian transpose?

Hermitian transpose is another name of the conjugate transpose matrix, mainly used on linear function spaces. Other names used: Hermitian conjugate, bedaggered matrix or transjugate.

### What is the adjoint matrix?

In English, the conjugate transposed matrix is sometimes erroneously called adjoint matrix but it is not the same matrix.

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