Tool to calculate adjoint matrix (or Hermitian transpose). The adjoint matrix is the transpose of the conjugate matrix of a matrix M.

Conjugate Transpose Matrix - dCode

Tag(s) : Mathematics, Algebra, Symbolic Computation

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Tool to calculate adjoint matrix (or Hermitian transpose). The adjoint matrix is the transpose of the conjugate matrix of a matrix M.

The conjugate transpose matrix is calculated for a matrix with complex elements. This is the transposed matrix of the conjugate matrix (or the conjugate matrix of the transposed matrix).

Consider \( M=[a_{ij}] \) a matrix, the conjugate transpose matrix is noted \( M^*=\overline{M}^T=\overline{M^T}=[\overline{a_{ij}}]^T \) (There is also a notation with a dagger †).

Example: $$ M=\begin{pmatrix} 2 & 1-i & 0 \\ 1 & 2+i & -i \end{pmatrix} \Rightarrow M^*= \begin{pmatrix} 2 & 1 \\ 1+i & 2-i \\ 0 & i \end{pmatrix} $$

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 called adjoint matrix but it is not the same as the generally defined adjoint matrix.

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