When the conjugate transpose of a complex square matrix is equal to the inverse of itself, then such matrix is called as unitary matrix. If Q is a complex square matrix and if it satisfies Qθ = Q-1 then such matrix is termed as unitary. Please note that Qθ and Q-1 represent the conjugate transpose and inverse of the matrix Q, respectively.
The conjugate transpose of a matrix ‘Q’ can also be written as 
.
Unitary matrix examples
The example of 2 x 2 unitary matrix is
![\[\begin{bmatrix} \frac{1+i}{2} & \frac{-1+i}{2}\\ \frac{1+i}{2} & \frac{1-i}{2} \end{bmatrix}\]](https://electricalvoice.com/wp-content/ql-cache/quicklatex.com-eeff1be55e56bf939240b6cf3f5d0657_l3.png "Rendered by QuickLaTeX.com")
The example of 3 x 3 unitary matrix is
![\[\frac{1}{2}\begin{bmatrix} 1 & -i & -1+i\\ i & 1 & 1+i\\ 1+i & -1+i & 0 \end{bmatrix}\]](https://electricalvoice.com/wp-content/ql-cache/quicklatex.com-195a128bf39252669641bbc86e740e6b_l3.png "Rendered by QuickLaTeX.com")
Note: If A is a unitary matrix then it will satisfy AAθ = I = AθA
