Special Orthogonal Matrix

A square matrix is a special orthogonal matrix if

(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] :=
    (Transpose[m] . m == IdentityMatrix @ Length @ m
&& Det[m] == 1)

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