Set of 2-Dimensional Indefinite Real Orthogonal Matrices is not Compact in Normed Real Square Matrix Vector Space
Theorem
Let $\struct {\R^{2 \times2}, \norm {\, \cdot \,}_\infty}$ be the normed real matrix vector space.
Let $\map O {1, 1} := \set {\mathbf R \in \R^{2 \times2} : \mathbf R^\intercal \mathbf J_{1,1} \mathbf R = \mathbf J_{1,1}}$ be the indefinite orthogonal group of degree $\paren {1, 1}$ over real numbers where:
- $\ds \mathbf J_{1,1} := \begin{bmatrix} 1 & 0\\ 0 & -1 \\ \end{bmatrix}$
Then $\map O {1, 1}$ is not a compact set in $\struct {\R^{2 \times 2}, \norm {\, \cdot \,}_\infty}$.
Proof
Let:
- $\begin{bmatrix} \map \cosh t& \map \sinh t\\ \map \sinh t & \map \cosh t \\ \end{bmatrix} := \map {\mathbf R} t$
We have that:
\(\ds \map {\mathbf R^\intercal} t \mathbf J_{1,1} \map {\mathbf R} t\) | \(=\) | \(\ds \begin{bmatrix} \map \cosh t & \map \sinh t \\ \map \sinh t & \map \cosh t \\ \end{bmatrix} \begin{bmatrix} 1 & 0\\ 0 & -1 \\ \end{bmatrix} \begin{bmatrix} \map \cosh t & \map \sinh t \\ \map \sinh t & \map \cosh t \\ \end{bmatrix}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \begin{bmatrix} \map {\cosh^2} t - \map {\sinh^2} t& 0 \\ 0 & \map {\sinh^2} t - \map {\cosh^2} t \\ \end{bmatrix}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \begin{bmatrix} 1 & 0 \\ 0 & -1 \\ \end{bmatrix}\) | Difference of Squares of Hyperbolic Cosine and Sine |
Hence, $\map {\mathbf R} t \in \map O {1, 1}$.
From difference of squares of hyperbolic cosine and sine:
\(\ds \size {\map \cosh t}\) | \(=\) | \(\ds \sqrt {\map {\sinh^2} t + 1}\) | ||||||||||||
\(\ds \) | \(>\) | \(\ds \size {\map \sinh t}\) |
The set of matrix elements constitutes a finite subset of the set of real numbers which is ordered.
By Finite Non-Empty Subset of Ordered Set has Maximal and Minimal Elements, there is a maximal element.
Consider the supremum norm of $\map {\mathbf R} t$:
\(\ds \norm {\map {\mathbf R} t}_\infty\) | \(=\) | \(\ds \max_{ \begin {split} & 1 \mathop \le i \mathop \le 2\\ & 1 \mathop \le j \mathop \le 2 \end {split} } \size {\map {r_{i j} } t}\) | Definition of Supremum Norm | |||||||||||
\(\ds \) | \(=\) | \(\ds \size {\map \cosh t}\) |
Furthermore:
- $\ds \lim_{t \mathop \to \infty} \map \cosh t = \infty$
Hence, $\map O {1,1}$ is not bounded.
By Heine-Borel theorem, $\map O {1,1}$ is not compact in $\struct {\R^{2 \times 2}, \norm {\, \cdot \,}_\infty}$.
$\blacksquare$
Sources
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 1.5$: Normed and Banach spaces. Compact sets