Matrix Groups

Set of all Invertible Matrices$GL_n(\mathbb{R})$

• Algebraic Structure: Forms a group under multiplication.
• Boundedness: Unbounded.
• Topology: Open in$M_n(\mathbb{R})$.
• Components: Has two connected components:$GL_n^+(\mathbb{R})$($det > 0$) and$GL_n^-(\mathbb{R})$($det < 0$).
• Center:$Z(GL_n(\mathbb{R})) = \{ \lambda I : \lambda \in \mathbb{R} \setminus \{0\} \}$.
• Functional Definition:$det^{-1}(\mathbb{R} \setminus \{0\})$.
• Connectivity: Not Connected.
• Determinant Range:$det(GL_n(\mathbb{R})) = \mathbb{R} \setminus \{0\}$.
• Density: Dense in$M_n(\mathbb{R})$.
  • Intersects all open sets.
  • Any singular matrix can be approximated by$GL_n(\mathbb{R})$.
  • Approximation Logic:$A + \delta I$has eigenvalues$\sigma(A) + \delta$.

Nilpotent Matrices

• Algebraic Structure: Not a group under addition or multiplication.
• Topology: Closed in$M_n(\mathbb{R})$.
• Functional Definition:$\bigcup_{k=1}^{n} f_k^{-1}(\{0\})$where$f_k(A) = A^k$.
• Spectrum: All eigenvalues are$0$.
• Trace/Det:$Trace(A) = 0$,$det(A) = 0$.
• Example: The set of strictly upper triangular matrices is a subset.
• Boundedness: Unbounded.
• Density: Not dense.
• Connectivity: Path Connected.
  • Note: Illustrated by points$A$and$B$where$A$and$B$are nilpotent, connected via a path through the origin$O$.

Idempotent Matrices

• Algebraic Structure: Not a subgroup under addition or multiplication.
• Connectivity: Not Connected.
• Determinant Properties: Determinant can take only 2 values:$\{0, 1\}$.
• Trace Properties:$Trace(\text{Idem. Mat.}) = \{0, 1, 2, \dots, n\}$.
• Eigenvalues:$\lambda \in \{0, 1\}$.
• Geometric Interpretation: Represents a projection operator$P: V \to V$.
• Rank-Trace Identity:$Rank(A) = Trace(A)$.
• Topology: Closed, defined by$(x^2 - A) = 0$or$(A^2 - A) = 0$.
• Boundedness: Unbounded.
  • Example:$A = \begin{bmatrix} 1 & n \\ 0 & 0 \end{bmatrix}$(shows it is not bounded as$n \to \infty$).
• Compactness: Not Compact (hence no).

Special Linear Group$SL_n(\mathbb{R})$

• Definition:$\{A \in GL_n(\mathbb{R}) : det A = 1\}$.
• Algebraic Structure: Subgroup of$GL_n(\mathbb{R})$.
• Boundedness: Unbounded.
• Center:$Z(SL_n(\mathbb{R})) = \{ \lambda I : \lambda^n = 1 \}$.
• Topology: Closed, defined by$det^{-1}(\{1\})$.
• Connectivity: Path Connected.
• Density: Not dense.

Orthogonal Matrices$O_n(\mathbb{R})$

• Definition:$\{A \in M_n(\mathbb{R}) : AA^T = I\}$.
• Properties: Rows and columns form an orthonormal basis for$\mathbb{R}^n$.
• Topology: Closed and bounded.
• Components: Two connected components ($SO_n$and$O_n \setminus SO_n$).
• Center:$Z(O_n(\mathbb{R})) = \{ \pm I \}$.
• Compactness: Compact.
• Functional Form:$(AA^T - I)^{-1}(\{0\})$.
• Connectivity: Not Connected.
• Determinant Range:$det(O_n(\mathbb{R})) = \{1, -1\}$.

Special Orthogonal Matrices$SO_n(\mathbb{R})$

• Definition:$\{A \in O_n(\mathbb{R}) : det A = 1\}$.
• Common Name: Called Rotation matrices.
• Case$n=2$: Has a specific rotation form.
• Case$n=3$:$1$is an eigenvalue.
• Eigenvalue Analysis:
  • If$n$is odd,$det(A - I) = det(A^T(A - I)) = det(I - A)$.
  • Since$det(I - A) = (-1)^n det(A - I)$, for$n=3$,$det(A - I) = 0$.
  • Therefore,$1$is an eigenvalue.
  • Except for trivial cases, eigenvalues are complex.
  • If eigenvalues are real, they must be$1$or$-1$; however,$-1$is not possible in$SO_n$for$n=3$without another$-1$to keep the determinant$1$.
• Rotation Axis: Eigenvector corresponding to the eigenvalue$1$is the axis of rotation.
• Fundamental Group: For$n \ge 3$,$\pi_1(SO_n(\mathbb{R})) \cong \mathbb{Z}_2$(Doubly connected).
• Topology: Path Connected and Compact.

Unitary Matrices$U_n(\mathbb{C})$

• Definition:$\{A \in M_n(\mathbb{C}) : A^*A = I\}$, where$A^*$is the conjugate transpose.
• Topology: They are Compact (closed and bounded in$\mathbb{C}^{n^2}$).
• Connectivity:$U_n(\mathbb{C})$is Connected, unlike$O_n(\mathbb{R})$.
• Determinant Range: The determinant is a complex number on the unit circle ($|det A| = 1$).

Special Unitary Group$SU_n(\mathbb{C})$

• Definition:$\{A \in U_n(\mathbb{C}) : det A = 1\}$.
• Topology: It is Compact and Simply Connected.
• Relation to Rotation:$SU(2)$is a "double cover" of$SO_3(\mathbb{R})$, meaning they are topologically related but$SU(2)$has no "holes" (it's simply connected).

Symmetric Matrices$S_n(\mathbb{R})$

• Definition:$\{A \in M_n(\mathbb{R}) : A = A^T\}$.
• Algebraic Structure: They form a vector subspace of$M_n(\mathbb{R})$, not a group under multiplication.
• Topology: Since it is a subspace, it is Closed, Unbounded, and Connected (specifically, it is homeomorphic to$\mathbb{R}^{n(n+1)/2}$).

Positive Definite Matrices$P_n$

• Definition: Symmetric matrices with all strictly positive eigenvalues ($x^T A x > 0$for all$x \neq 0$).
• Topology: They form an Open Cone within the space of symmetric matrices.
• Connectivity: They are Connected and Convex.
• Closure: The closure of$P_n$is the set of positive semi-definite matrices (eigenvalues$\ge 0$), which is a Closed set.

Hermitian Matrices$H_n(\mathbb{C})$

• Definition:$\{A \in M_n(\mathbb{C}) : A = A^*\}$.
• Key Property: Like symmetric matrices, all eigenvalues of a Hermitian matrix are real.
• Topology: They form a real vector space and are Connected and Closed.