In the context of the general linear group $G{L}_n(\mathbb{R})$, which consists of all $n\times n$ invertible matrices with real entries, a path is typically a continuous function from the interval $[0,1]$ to $G{L}_n(\mathbb{R})$.

Formally, a path in $G{L}_n(\mathbb{R})$ is a continuous map:

$$\gamma :[0,1]\to G{L}_n(\mathbb{R})$$

This means for each $t\in [0,1]$, $\gamma (t)$ is an $n\times n$ invertible matrix, and the function $\gamma$ is continuous with respect to the topology on $G{L}_n(\mathbb{R})$ induced by the standard topology on ${\mathbb{R}}^{n\times n}$.

Examples

1. Straight Line Path: Consider the matrices $A,B\in G{L}_n(\mathbb{R})$. A simple example of a path between $A$ and $B$ is given by:

   $$\gamma (t)=(1-t)A+tB\text{for}t\in [0,1]$$

   provided that $\gamma (t)\in G{L}_n(\mathbb{R})$ for all $t\in [0,1]$.

2. Exponential Map: Given a matrix $A\in G{L}_n(\mathbb{R})$, a common path starting from the identity matrix $I$ is:

   $$\gamma (t)=\exp (t\log (A))\text{for}t\in [0,1]$$

   Here, $\log (A)$ is the matrix logarithm of $A$ and $\exp$ is the matrix exponential. This ensures that $\gamma (0)=I$ and $\gamma (1)=A$.

Properties

• Homotopy: Two paths ${\gamma }_0$ and ${\gamma }_1$ in $G{L}_n(\mathbb{R})$ are said to be homotopic if there exists a continuous map $H:[0,1]\times [0,1]\to G{L}_n(\mathbb{R})$ such that:

  $$H(t,0)={\gamma }_0(t),H(t,1)={\gamma }_1(t),H(0,s)=I,H(1,s)=A$$

  for all $t,s\in [0,1]$.

• Fundamental Group: The set of all loops (paths that start and end at the same point) in $G{L}_n(\mathbb{R})$ based at the identity matrix, up to homotopy, forms the fundamental group ${\pi }_1(G{L}_n(\mathbb{R}),I)$. For $n\ge 2$, this fundamental group is known to be isomorphic to $\mathbb{Z}/2\mathbb{Z}$, reflecting the fact that $G{L}_n(\mathbb{R})$ has two connected components: matrices with positive determinant and matrices with negative determinant. For $n=1$, ${\pi }_1(G{L}_1(\mathbb{R}),I)$ is isomorphic to $\mathbb{Z}$.

Understanding paths in $G{L}_n(\mathbb{R})$ is essential in various areas of mathematics, including differential topology, algebraic geometry, and the study of Lie groups and their representations.