Real Orthogonal Matrices

Theorem: Orthonormal Bases from an Orthogonal Matrix

If $A\in {\mathbb{R}}^{n\times n}$ is a real orthogonal matrix, then:

• the columns of $A$ form an Inner Product Spaces of the vector space ${\mathbb{R}}^n$;

• the columns of its transpose $A^{\mathsf{T}}$ also form an Inner Product Spaces of the vector space ${\mathbb{R}}^n$.

PROOF

TODO

Theorem: Length Preservation of Real Orthogonal Matrices

Multiplying a column vector $\vec{v}\in {\mathbb{R}}^n$ by a real orthogonal matrix $A\in {\mathbb{R}}^{n\times n}$ preserves its length.

$$||A\vec{v}||=||\vec{v}||$$

PROOF

TODO