Product of Orthogonal Matrices is Orthogonal Matrix

Theorem

Let $\mathbf P$ and $\mathbf Q$ be orthogonal matrices.

Let $\mathbf P \mathbf Q$ be the (conventional) matrix product of $\mathbf P$ and $\mathbf Q$.

Then $\mathbf P \mathbf Q$ is an orthogonal matrix.

Proof

From Determinant of Orthogonal Matrix is Plus or Minus One and Matrix is Invertible iff Determinant has Multiplicative Inverse it follows that both $\mathbf P$ and $\mathbf Q$ are invertible.

Thus:

\(\ds \paren {\mathbf P \mathbf Q}^{-1}\)

\(=\)

\(\ds \mathbf Q^{-1} \mathbf P^{-1}\)

Inverse of Matrix Product

\(\ds \)

\(=\)

\(\ds \mathbf Q^\intercal \mathbf P^\intercal\)

Definition of Orthogonal Matrix

\(\ds \)

\(=\)

\(\ds \paren {\mathbf P \mathbf Q}^\intercal\)

Transpose of Matrix Product

Hence the result, by definition of orthogonal matrix.

$\blacksquare$

Sources

• 1980: A.J.M. Spencer: Continuum Mechanics ... (previous) ... (next): $2.1$: Matrices
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): orthogonal matrix