Definition

A n×nn \times n matrix 𝐀\mathbf{A} is an orthogonal matrix if

𝐀𝐀𝖳=𝐀𝖳𝐀=I\mathbf{AA}^{\sf T} = \mathbf{A}^{\sf T}\mathbf{A} = I

where 𝐀𝖳\mathbf{A}^{\sf T} is the transpose of 𝐀\mathbf{A} and 𝐈\mathbf{I} is the identity matrix.

(a real square matrix with columns and rows that are orthonormal vectors)

Orthogonal matrices are always invertible,

𝐀−1=𝐀𝖳\mathbf{A}^{-1} = \mathbf{A}^{\sf T}

The rows of an orthogonal matrix form an orthonormal basis.

References:

  1. https://mathworld.wolfram.com/OrthogonalMatrix.html
  2. https://www.ucl.ac.uk/~ucahmdl/LessonPlans/Lesson10.pdf