Orthogonal Matrices

Definition: Orthogonal Matrix

An orthogonal matrix is a square matrix whose inverse is its own transpose.

$$M^{-1}=M^{\mathsf{T}}$$

Theorem: Product of Orthogonal Matrices

The product of two orthogonal matrices is also an orthogonal matrix.

PROOF

$$(AB{)}^{\mathsf{T}}(AB)=B^{\mathsf{T}}{A}^{\mathsf{T}}AB=B^{\mathsf{T}}B=I_n$$

Theorem: Determinant of an Orthogonal Matrix

The determinant of an orthogonal matrix is either $+1$ or $-1$.

PROOF

From $A^{\mathsf{T}}A=I_n$ we have

$$\det (A^{\mathsf{T}}A)=\det (I_n)$$ $$\det (A^{\mathsf{T}})\det (A)=1$$ $$\det (A{)}^2=1⟹\det (A)=\pm 1$$