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

Conjugate Transpose Matrix - dCode

Tag(s) : Matrix

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

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 $.

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.

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.

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

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*Conjugate Transpose Matrix* on dCode.fr [online website], retrieved on 2024-09-10,

conjugate,transpose,complex,matrix,asterisk,dagger

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