# Statement(s) > [!NOTE] Statement 1 (Normal Form of Orthogonal Matrix) > Let $L:\mathbb{R}^{n} \to \mathbb{R}^{n}$ be a [[Isometry group of real n-space with standard metric|linear isometry]] given by an [[Orthogonal endomorphisms of Euclidean spaces|orthogonal matrix]] $A$. Then there exists an [[Orthonormal Subset of Real n-Space|orthonormal]] basis of $\mathbb{R}^n$ in which the matrix of $L$ is $\begin{pmatrix} > I_{k} \\ > &- I_{m} \\ > & & B_{1} \\ > & & & \ddots \\ > & & & & B_{l} > \end{pmatrix} \quad \text{where} \quad B_{i} = \begin{pmatrix} > \cos \theta_{i} & -\sin \theta_{i} \\ > \sin \theta_{i} & \cos \theta_{i} > \end{pmatrix}.$where $k,m,l\in \mathbb{N}$ such that $k+2l+m=n$ and $I_{k}$ is the $k\times k$ [[Real Identity Matrix|identity matrix]]. # Proof(s) ###### Proof of Statement 1 (MA243): We proceed by induction on $n$. It follows from [[General form of Plane Isometries]] that this statement holds true for $n=2$, since either $A$ is a [[Plane Rotation Matrix|rotation matrix]] which already has the required form or $A$ is a [[Plane Reflection Matrix|reflection matrix]] in the line at angle $\theta$ above the $x$-axis which has eigenvalues $\pm 1$ and orthogonal eigenvectors $(\cos\theta,\sin \theta)$ and $(\sin\theta,-\cos\theta)$. In the latter case, $L$ has matrix $\text{diag}(1,-1)$ wrt the eigenvectors basis. Suppose $n>2.$