Conjugate Transpose Matrix

Question 1

What is a matrix conjugate transpose? (Definition)

Answer

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 $.

Question 2

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

Answer

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.

Question 3

What is the hermitian transpose?

Answer

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.

Question 4

What is the adjoint matrix?

Answer

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

Conjugate Transpose Matrix