Conjugate of a Matrix
If $A$ is a matrix with complex elements, the conjugate of $A$, denoted as $\stackrel{\u203e}{A}$, is obtained by taking the conjugate of each element within the matrix.
Representation:
If $A$ is an $m\times n$ matrix with complex elements ${a}_{ij}$, then the conjugate of $A$, $\stackrel{\u203e}{A}$, is represented as a matrix with elements $\stackrel{\u203e}{{a}_{ij}}$.
Example:
Let's consider a matrix $A$ with complex elements:
The conjugate of $A$, denoted as $\stackrel{\u203e}{A}$, would be:
Properties:

$\stackrel{\u203e}{\stackrel{\u203e}{A}}=A$: Conjugating a matrix twice brings it back to the original matrix.

$\stackrel{\u203e}{(kA)}=\stackrel{\u203e}{k}\cdot \stackrel{\u203e}{A}$: Conjugate of a scalar multiplied by a matrix is equal to the conjugate of the scalar multiplied by the conjugate of the matrix.

$\stackrel{\u203e}{(A+B)}=\stackrel{\u203e}{A}+\stackrel{\u203e}{B}$: Conjugate of the sum of matrices is equal to the sum of their conjugates.

$\stackrel{\u203e}{(AB)}=\stackrel{\u203e}{A}\cdot \stackrel{\u203e}{B}$: Conjugate of the product of two matrices is equal to the product of their conjugates in the same order.
Applications:

Quantum Mechanics: Conjugate transpose matrices are significant in quantum mechanics, especially in representing quantum states and operations.

Signal Processing: In complex signal processing applications where complex numbers are involved, conjugates are used for various transformations and operations.