Product of orthogonal matrices

Definition

A real square matrix Q is orthogonal if QQᵀ = QᵀQ = I, i.e. Qᵀ = Q⁻¹. (Its rows, and its columns, form an orthonormal set, and |Q| = ±1.) Example: the rotation matrix [cos θ −sin θ; sin θ cos θ].

Proof

Let A and B be orthogonal matrices of the same order n, so

AAᵀ = AᵀA = I   and   BBᵀ = BᵀB = I

Use the reversal law (AB)ᵀ = BᵀAᵀ. Then

(AB)(AB)ᵀ = (AB)(BᵀAᵀ)
          = A (BBᵀ) Aᵀ
          = A I Aᵀ
          = AAᵀ
          = I

Similarly,

(AB)ᵀ(AB) = BᵀAᵀAB = Bᵀ(AᵀA)B = BᵀIB = BᵀB = I

Both products equal I, so AB is orthogonal. ∎

Alternative (using inverses): Orthogonality gives |A| = ±1 and |B| = ±1, so |AB| = ±1 ≠ 0, and (AB)⁻¹ = B⁻¹A⁻¹ = BᵀAᵀ = (AB)ᵀ.

Example. A = [0 1; −1 0] and B = [0 −1; −1 0] are both orthogonal, and AB = [−1 0; 0 1], which satisfies (AB)(AB)ᵀ = I.