Definition:Complex Conjugate of Matrix

Definition

Let $\mathbf A = \sqbrk \alpha_{m n}$ be an $m \times n$ matrix over the complex numbers $\C$.

Let $\overline {\mathbf A}$ denote the matrix formed from $\mathbf A$ by replacing each entry of $\mathbf A$ with its complex conjugate:

$\overline {\mathbf A} = \sqbrk \beta_{m n}: \forall i \in \set {1, 2, \ldots, m}, j \in \set {1, 2, \ldots, n}: \beta_{i j} = \overline {\alpha_{i j} }$

where $\overline {\alpha_{i j} }$ denotes the complex conjugate of $\alpha_{j i}$.

Then $\overline {\mathbf A}$ is the complex conjugate of $\mathbf A$.

Also denoted as

The complex conjugate of a matrix $\mathbf A$ can also be seen denoted by $\mathbf A^*$.

Also see

• Definition:Hermitian Conjugate

• Results about complex conjugates can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): complex conjugate: 2.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): complex conjugate: 2.