Product of Proper Orthogonal Matrices is Proper Orthogonal Matrix
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Theorem
Let $\mathbf P$ and $\mathbf Q$ be proper orthogonal matrices.
Let $\mathbf {P Q}$ be the (conventional) matrix product of $\mathbf P$ and $\mathbf Q$.
Then $\mathbf {P Q}$ is a proper orthogonal matrix.
Proof
By definition, $\mathbf {P Q}$ is a proper orthogonal matrix if and only if it is an orthogonal matrix with a determinant of $1$.
From Product of Orthogonal Matrices is Orthogonal Matrix, $\mathbf {P Q}$ is an orthogonal matrix.
By definition, $\mathbf P$ and $\mathbf Q$ both have a determinant of $1$.
From Determinant of Matrix Product:
- $\map \det {\mathbf {P Q} } = \map \det {\mathbf P} \map \det {\mathbf Q}$
Thus:
- $\map \det {\mathbf {P Q} } = 1$
Hence the result.
$\blacksquare$