ORTHOGONAL MATRIX

An orthogonal matrix (or orthonormal matrix), is a real square matrix whose columns and rows are orthonormal vectors.

One way to express this is

$$Q^{T}Q=Q{Q}^T=I$$

This leads to the equivalent characterization: a matrix $Q$ is orthogonal if its transpose is equal to its inverse:

$QT=Q^{-1}$

The determinant of any orthogonal matrix is either $+1$ or $-1$. As a linear transformation, an orthogonal matrix preserves the inner product of vectors, and therefore acts as an isometry of Euclidean space, such as a rotation, reflection or rotoreflection. In other words, it is a unitary transformation.

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