What is Unitary Matrix? Example

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 Q^* or (\overline{Q})^T


Unitary matrix examples

The example of 2 x 2 unitary matrix is

    \[\begin{bmatrix} \frac{1+i}{2} & \frac{-1+i}{2}\\ \frac{1+i}{2} & \frac{1-i}{2} \end{bmatrix}\]

The example of 3 x 3 unitary matrix is

    \[\frac{1}{2}\begin{bmatrix} 1 & -i & -1+i\\ i & 1 & 1+i\\ 1+i & -1+i & 0 \end{bmatrix}\]

Note: If A is a unitary matrix then it will satisfy AAθ = I = AθA

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