Conjugation

Definition: The complex conjugate of a matrix $A = (a_{ij})$ with complex entries is the matrix $\overline{A}$ obtained by taking the complex conjugate of each entry:

$$(\overline{A})_{ij} = \overline{a_{ij}}$$

where $\overline{a+bi} = a-bi$.

Example:
$$A = \begin{pmatrix} 1+i & 2-3i \\ 4 & 5i \end{pmatrix}, \quad \overline{A} = \begin{pmatrix} 1-i & 2+3i \\ 4 & -5i \end{pmatrix}$$

Properties:

• $\overline{\overline{A}} = A$
• $\overline{A + B} = \overline{A} + \overline{B}$
• $\overline{AB} = \overline{A}\,\overline{B}$
• $\overline{cA} = \overline{c}\,\overline{A}$
• $\det(\overline{A}) = \overline{\det(A)}$

Conjugation is often combined with transposition to form the conjugate transpose (adjoint): $A^* = (\overline{A})^T$.