Special Orthogonal Matrix

A special orthogonal matrix is a square matrix with real entries satisfying

(1)

where is the identity matrix, and the determinant satisfies

(2)

The first condition means that is an orthogonal matrix, and the second restricts the determinant to (while a general orthogonal matrix may have determinant or ). For example,

(3)

is a special orthogonal matrix since

(4)

and its determinant is . A matrix can be tested to see if it is a special orthogonal matrix using the Wolfram Language code

SpecialOrthogonalQ[m_List?MatrixQ] :=
 (Dimensions[m] == {Length[m], Length[m]}&&
 MatrixQ[m, (Element[#, Reals] === True)&]&&
 Transpose[m] . m == IdentityMatrix[Length[m]]&&
 Det[m] == 1)

The special orthogonal matrices are closed under matrix multiplication and the matrix inverse operation, and therefore form a matrix group called the special orthogonal group .