The compactness of the matrix groups $S{L}_n$, $G{L}_n$, $S{O}_n$, $O_n$, $S{U}_n$, and $U_n$ varies depending on the specific group. Here’s a detailed explanation of their compactness:

1. Special Linear Group $S{L}_n(\mathbb{R})$

• Definition: $S{L}_n(\mathbb{R})$ is the group of $n\times n$ real matrices with determinant 1.
• Compactness:
  • For $n\ge 2$: $S{L}_n(\mathbb{R})$ is not compact. It is a closed subset of the general linear group $G{L}_n(\mathbb{R})$, which is open in ${\mathbb{R}}^{n^2}$. While $S{L}_n(\mathbb{R})$ is closed, it is not bounded. The reason for this is that matrices in $S{L}_n(\mathbb{R})$ can have arbitrarily large norms. For example, diagonal matrices with entries increasing without bound (while maintaining determinant 1) show that $S{L}_n(\mathbb{R})$ is unbounded in ${\mathbb{R}}^{n^2}$.
  • For $n=1$: $S{L}_1(\mathbb{R})$ consists of the single matrix $[1]$, which is compact because it is a finite set.

2. General Linear Group $G{L}_n(\mathbb{R})$

• Definition: $G{L}_n(\mathbb{R})$ is the group of $n\times n$ invertible real matrices.
• Compactness:
  • $G{L}_n(\mathbb{R})$ is not compact. It is an open subset of ${\mathbb{R}}^{n^2}$, and hence it is unbounded. Specifically, matrices in $G{L}_n(\mathbb{R})$ can have entries with arbitrarily large magnitudes, leading to a lack of boundedness. For instance, one can construct matrices with entries growing without bound, which would still be invertible.

3. Special Orthogonal Group $S{O}_n(\mathbb{R})$

• Definition: $S{O}_n(\mathbb{R})$ is the group of $n\times n$ orthogonal matrices with determinant 1.
• Compactness:
  • For $n\ge 2$: $S{O}_n(\mathbb{R})$ is compact. It is a closed subset of $O_n(\mathbb{R})$, which is the group of all orthogonal matrices (with determinant $\pm 1$). Since $O_n(\mathbb{R})$ is compact and $S{O}_n(\mathbb{R})$ is a closed subgroup of it, $S{O}_n(\mathbb{R})$ inherits compactness. Additionally, $S{O}_n(\mathbb{R})$ is bounded because all orthogonal matrices have norm 1, and it can be shown to be a compact Lie group.

4. Orthogonal Group $O_n(\mathbb{R})$

• Definition: $O_n(\mathbb{R})$ is the group of $n\times n$ orthogonal matrices with determinant $\pm 1$.
• Compactness:
  • $O_n(\mathbb{R})$ is compact. It is a closed subset of ${\mathbb{R}}^{n^2}$ and is bounded. The compactness can be seen from the fact that orthogonal matrices satisfy $A^{T}A=I$, which restricts their entries, and hence, they form a bounded set in ${\mathbb{R}}^{n^2}$. This group includes matrices with determinant 1 (forming $S{O}_n(\mathbb{R})$) and those with determinant -1, but remains compact as it is the union of these two compact sets.

5. Special Unitary Group $S{U}_n(\mathbb{C})$

• Definition: $S{U}_n(\mathbb{C})$ is the group of $n\times n$ unitary matrices with determinant 1.
• Compactness:
  • $S{U}_n(\mathbb{C})$ is compact. It is a closed and bounded subset of the unitary group $U_n(\mathbb{C})$, which consists of all unitary matrices. The compactness comes from the fact that $S{U}_n(\mathbb{C})$ is a closed subgroup of $U_n(\mathbb{C})$ (which is compact). The condition that unitary matrices have norm 1 ensures boundedness, and the determinant condition does not affect the compactness.

6. Unitary Group $U_n(\mathbb{C})$

• Definition: $U_n(\mathbb{C})$ is the group of $n\times n$ unitary matrices.
• Compactness:
  • $U_n(\mathbb{C})$ is compact. It is a closed subset of ${\mathbb{C}}^{n^2}$, and unitary matrices have entries satisfying certain norm conditions, which ensures boundedness. The compactness can be attributed to the fact that unitary matrices are defined by the equation $U^{\ast }U=I$, where $U^{\ast }$ is the conjugate transpose, enforcing a bounded structure.

Summary

• $S{L}_n(\mathbb{R})$: Not compact for $n\ge 2$, compact for $n=1$.
• $G{L}_n(\mathbb{R})$: Not compact; unbounded.
• $S{O}_n(\mathbb{R})$: Compact for $n\ge 2$.
• $O_n(\mathbb{R})$: Compact.
• $S{U}_n(\mathbb{C})$: Compact.
• $U_n(\mathbb{C})$: Compact.