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

$$ Q^TQ=QQ^T=I $$

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.