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Consider the general linear group $$GL(n,\mathbb R)=\{g\in {\mathbb R}^{n\times n}\mid\det(g)\neq 0\}$$ Prove that the derivative of the function $f=\det:{\mathbb R}^{n\times n}\to\mathbb R$ is given by $$df(g)v=\det(g)\operatorname{trace}(g^{-1}v)$$ for every $g\in GL(n,\mathbb R)$ and every $v\in {\mathbb R}^{n\times n}$. Deduce that the special linear group $$SL(n,\mathbb R):=\{g\in GL(n,\mathbb R)\mid\det(g)=1\}$$ is a smooth submainfold of ${\mathbb R}^{n\times n}$

I have a bunch of questions to ask before I can solve it: Is the dimension of $GL(n,\mathbb R)$ $n^2$? And what is the dimension of the special linear group? What are the basis vectors of the special linear group? Is the general linear space is a metric space? I was trying to compute $df(g)v$, and by definition, $df(g)v=\lim_{t\to 0}\dfrac{\det(g+tv)-\det(g)}{t}$. But how can I expand $ \det(g+tv)$?

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  • $\begingroup$ What do you meant by dimension? $\endgroup$ – copper.hat Dec 17 '14 at 20:04
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    $\begingroup$ The general linear group is not a vector space, because it doesn't contain the zero operator. Thus, it does not have a dimension. $\endgroup$ – Nishant Dec 17 '14 at 20:06
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    $\begingroup$ @Nishant I think the OP means dimension as a manifold. $\endgroup$ – André 3000 Dec 17 '14 at 20:07
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    $\begingroup$ The derivative is straightforward to compute for the matrix $I$. $\endgroup$ – copper.hat Dec 17 '14 at 20:09
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    $\begingroup$ For the determinant question, this post is helpful. $\endgroup$ – André 3000 Dec 17 '14 at 20:11
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Some answers/hints:

  1. Yes the dimension of $GL(n,\mathbb{R})$ is $n^2$. In fact it's the open subset of $\mathbb{R}^{n^2}$ on which the determinant (which is continuous) is nonzero.

  2. The special linear group is the subset of the general linear group on which the determinant is one. Neither of these is itself a vector space, so talking about basis vectors doesn't make sense. Here's a hint: If we have $f: M\rightarrow \mathbb{R}$ and consider $N = f^{-1}(\{c\})$, for some $c\in \mathbb{R}$, when is $N$ a manifold? What is its dimension when it is a manifold?

  3. There are various ways to find the derivative of $\det$. Try writing it out in coordinates and using multilinearity of the determinant. (This is easiest when $g=I$, so start there. If you like you can deduce the whole formula from its value at $g=I$ by using the matrix multiplication map $GL_n \times GL_n \rightarrow GL_n$ and the chain rule.)

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  • $\begingroup$ Is $f$ the one appears in my question? If it's not, then is it smooth? $\endgroup$ – pxc3110 Dec 17 '14 at 22:18
  • $\begingroup$ Yes, take $f$ smooth. (The hint is about general such $f$. In your problem $M = GL(n, \mathbb{R})$ and $f = \det$, with $c = 1$ and $N = SL(n, \mathbb{R})$.) $\endgroup$ – aes Dec 17 '14 at 23:45
  • $\begingroup$ I think $g=I$ is not the easiest case. In contrast, I found it easiest to prove when letting $v=g$. So now I'm trying to show that $tr(g^{-1}v)=\lim_{t\to 0} \frac{\det (I+tg^{-1}v)-1}{t}$ $\endgroup$ – pxc3110 Dec 18 '14 at 13:50
  • $\begingroup$ That's equivalent to the $g = I$ case, with $g^{-1}v$ in place of $v$. $\endgroup$ – aes Dec 18 '14 at 17:49

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