An [[Orthogonal Matrix]] is a type of [[Matrix|Square Matrix]] such that the corresponding [[Linear Transformation]] perserves angles and does not scale. A [[Matrix]] can only be [[Orthogonal]] [[Bijective|if and only if]] its [[Inverse Matrices|Inverse]] equals its [[Matrix Transpose|Transpose]]. $\huge \begin{align} A&\in M_{n\times n}\\ A^{-1} &= A^{T}\\ \end{align} $ [[Universal Quantifier|For any]] two [[Vector|Vectors]] $\vec u, \vec v \in \R^{n}$, the [[Vector Magnitude|Magnitudes]] of each [[Vector]] is unchanged when [[Matrix Product|multiplied]] by the matrix, and the angle $\theta$ between $\vec u$ and $\vec v$ is equal to the angle between $A\vec u$ and $A\vec v$. $\large\begin{align} \forall \vec v, \vec u &\in \R^n\\ \lvert \lvert \vec u \rvert \rvert &= \lvert \lvert A\vec u \rvert \rvert \\ \lvert \lvert \vec v \rvert \rvert &= \lvert \lvert A\vec v \rvert \rvert \\ \theta_{\angle \vec u, \vec v} &= \theta_{\angle A\vec u, A\vec v} \\ \end{align} $ A more consise way of expressing this relationship is to say that the [[Dot Product]] between the two [[Vector|Vectors]] is equal to the [[Dot Product]] between the transformed [[Vector|Vectors]]. $\huge \vec u \cdot \vec v = A\vec u \cdot A\vec v $ ### [[Gram Matrix]] The [[Gram Matrix]] of an [[Orthogonal Matrix]] is the [[Identity Matrix]]. >[!tldr] Proof for the [[Gram Matrix]] being the [[Identity]]. >$\large >\begin{align} >G_{A} &= AA^T \\ >&= AA^{-1} \\ >&= I >\end{align} >$ ### Column Vectors [[Universal Quantifier|For any]] [[Orthogonal Matrix]] $A \in M_{n\times n}$, all [[Vector|Column Vectors]], $\vec A_{1}, \dots, \vec A_{n}$, must be a [[Unit Vector]] and must be [[Orthogonal]] to all other [[Vector|Column Vectors]]. ### [[Determinant]] The [[Determinant]] of a [[Orthogonal Matrix]] must be $1$ or $-1$ (note: this is not a [[Bijective]] [[Proposition|Statement]]). ### [[Eigenvector|Eigenvectors]] [[Universal Quantifier|For All]] [[Eigenvalue|Eigenvalues]] of a [[Orthogonal Matrix]] $A$, the [[Absolute Value]] of the [[Eigenvalue]] must be $1$. $\huge \, \lvert \lambda_i \rvert = 1 $