The Unapologetic Mathematician

General Linear Groups are Lie Groups

One of the most important examples of a Lie group we’ve already seen: the general linear group $GL(V)$ of a finite dimensional vector space $V$. Of course for the vector space $\mathbb{R}^n$ this is the same as — or at least isomorphic to — the group $GL_n(\mathbb{R})$ of all invertible $n\times n$ real matrices, so that’s a Lie group we can really get our hands on. And if $V$ has dimension $n$, then $V\cong\mathbb{R}^n$, and thus $GL(V)\cong GL_n(\mathbb{R})$.

So, how do we know that it’s a Lie group? Well, obviously it’s a group, but what about the topology? The matrix group $GL_n(\mathbb{R})$ sits inside the algebra $M_n(\mathbb{R})$ of all $n\times n$ matrices, which is an $n^2$-dimensional vector space. Even better, it’s an open subset, which we can see by considering the (continuous) map $\mathrm{det}:M_n(\mathbb{R})\to\mathbb{R}$. Since $GL_n(\mathbb{R})$ is the preimage of $\mathbb{R}\setminus\{0\}$ — which is an open subset of $\mathbb{R}$$GL_n(\mathbb{R})$ is an open subset of $M_n(\mathbb{R})$.

So we can conclude that $GL_n(\mathbb{R})$ is an open submanifold of $M_n$, which comes equipped with the standard differentiable structure on $\mathbb{R}^{n^2}$. Matrix multiplication is clearly smooth, since we can write each component of a product matrix $AB$ as a (quadratic) polynomial in the entries of $A$ and $B$. As for inversion, Cramer’s rule expresses the entries of the inverse matrix $A^{-1}$ as the quotient of a (degree $n-1$) polynomial in the entries of $A$ and the determinant of $A$. So long as $A$ is invertible these are two nonzero smooth functions, and thus their quotient is smooth at $A$.

June 9, 2011 -