Orthogonal Matrix
An orthogonal matrix $Q$ is a square matrix whose transpose is equal to its inverse: ${Q}^{T}={Q}^{1}$
 Properties:
 For real matrices, being orthogonal is equivalent to preserving lengths and angles, similar to a rotation or reflection.
 For complex matrices, it satisfies $Q{Q}^{\ast}={Q}^{\ast}Q=I$, where ${Q}^{\ast}$ is the conjugate transpose of $Q$ and $I$ is the identity matrix.
 Orthogonal matrices have orthonormal columns and rows.
Example of an Orthogonal Matrix:
$$Q=\frac{1}{\sqrt{2}}\left[\begin{array}{cc}{\textstyle 1}& {\textstyle 1}\\ {\textstyle 1}& {\textstyle 1}\end{array}\right]$$
In this case, ${Q}^{T}={Q}^{1}$, which makes it an orthogonal matrix.
Properties of Orthogonal Matrices:

Transpose and Inverse: The transpose of an orthogonal matrix is equal to its inverse: ${Q}^{T}={Q}^{1}$.

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

Preservation of Length and Angles: Orthogonal matrices preserve lengths of vectors and angles between vectors, maintaining orthogonality.