Unitary Matrix
A square matrix $U$ is unitary if its conjugate transpose is equal to its inverse: ${U}^{\u2020}={U}^{1}$
 Properties:
 In a real matrix, being unitary is equivalent to being orthogonal.
 In a complex matrix, it satisfies $U{U}^{\u2020}={U}^{\u2020}U=I$, where $I$ is the identity matrix.
 Unitary matrices preserve the inner product of vectors, maintaining lengths and angles between vectors.
Example of a Unitary Matrix:
$$U=\frac{1}{\sqrt{2}}\left[\begin{array}{cc}{\textstyle 1}& {\textstyle 1}\\ {\textstyle 1}& {\textstyle 1}\end{array}\right]$$
In this case, ${U}^{\u2020}={U}^{1}$, which makes it a unitary matrix.
Properties of Unitary Matrices:

Conjugate Transpose: The conjugate transpose of a unitary matrix is its inverse: ${U}^{\u2020}={U}^{1}$.

Orthonormal Columns: The columns of a unitary matrix form an orthonormal set of vectors, meaning they are pairwise orthogonal and have unit length.

Preservation of Inner Product: $U$ preserves the inner product of vectors $v$ and $w$: $\u27e8Uv,Uw\u27e9=\u27e8v,w\u27e9$.