How many pairwise nonsimilar 3x3 diagonal orthogonal matrice

How many pairwise non-similar 3x3 diagonal orthogonal matrices are there?

Solution

A square matrix can represent any linear vector translation. Sometimes we want to constrain the elements of the matrix so that it represents a pure solid body rotation. A matrix representation of a rotation therefore contains redundant information, a 3D rotation has 3 degrees of freedom but a 3×3 matrix has 9 scalar values.

matrix A is orthogonal if:

All Orthogonal Matrices have determinants of 1 or -1 and all rotation matrices have determinants of 1

For example mode of R = cos(a)² + sin(a)² = 1

If we have a 3x3 matrix, how can we check if it represents an orthogonal matrix? Well we could check the things mentioned above, such as,