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What is a Unitary Matrix?
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Consider DFT matrix for $N=4,A=\dfrac{1}{√4}\begin{bmatrix}1&1&1&1 \\ 1&-j&-1&j \\ 1&-1&1&-1 \\ 1&j&-1&-j\end{bmatrix}$

If AA*=I

Then A is a unitary matrix

For N=4,

AA*$= \dfrac{1}{√4} \begin{bmatrix}1&1&1&1 \\ 1&-j&-1&j \\ 1&-1&1&-1 \\ 1&j&-1&-j \end{bmatrix} \dfrac{1}{√4} \begin{bmatrix}1&1&1&1 \\ 1&j&-1&-j \\ 1&-1&1&-1 \\ 1&-j&-1&j \end{bmatrix} \\ = \dfrac14 \begin{bmatrix}4&0&0&0 \\ 0&4&0&0 \\ 0&0&4&0 \\ 0&0&0&4 \end{bmatrix} \\ = \begin{bmatrix}1&0&0&0 \\ 0&1&0&0 \\ 0&0&1&0 \\ 0&0&0&1 \end{bmatrix}$

AA*= I

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