Special Orthogonal Matrix

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

(1)

where is a positive integer, is the transpose of , 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 .

Its matrix inverse is . The linear transformation preserves the standard inner product on and is orientation-preserving. In two dimensions, every special orthogonal matrix is a rotation matrix of the form

(5)

where is a real number (Ng et al. 2021, pp. 54-56). Acting on column vectors, it rotates the plane counterclockwise through angle . The example above has .

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 .