# 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:

$Q=\frac{1}{\sqrt{2}}\left[\begin{array}{cc}1& -1\\ 1& 1\end{array}\right]$

In this case, ${Q}^{T}={Q}^{-1}$, which makes it an orthogonal matrix.

### Properties of Orthogonal Matrices:

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

2. 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.

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