Orthogonality

In linear algebra, we say that vectors are orthogonal if their dot product is equal to 0.

If we say that a set is orthonormal if all the vectors in the set are mutually orthogonal and have a norm of 1. We similarly define an orthonormal basis of a subspace of $V$ is an orthonormal set that is a basis of $V$.

Matrices

An orthogonal matrix $T:{\mathbb{R}}^n\to {\mathbb{R}}^n$ is if $\Vert T(x)\Vert =\Vert \vec{x}\Vert$ for all $\vec{x}\in {\mathbb{R}}^n$ and if and only if $T({\vec{e}}_1),T({\vec{e}}_2),\dots ,T({\vec{e}}_n)$ form an orthonormal basis for ${\mathbb{R}}^n$. This happens when the columns of $A$ are orthonormal.

Additionally, it is orthogonal if and only if $T(\vec{v})\cdot T(\vec{w})=\vec{v}\cdot \vec{w}$ for all $\vec{v},\vec{w}\in {\mathbb{R}}^n$, and if and only if $A^{T}A=I_n$ or equivalently if $A^{-1}=A^T$.

Note also:

$$\text{det}(I_n)=1=\text{det}(A^{T}A)=\text{det}(A)$$

Theorems

The possible real eigenvalues of an orthogonal matrix are 1 and -1.