Let $F:Tr(n,\mathbb{R})\cap GL_n(\mathbb{R})\rightarrow Tr(n,\mathbb{R})\cap GL_n(\mathbb{R})$ be the map which sends a matrix $A$ to its inverse $A^{-1}$. If we consider $F$ as a function from $(\mathbb{R}^\times)^n\times \mathbb{R}^{n(n-1)/2}$ into itself, is there a formula for the jacobian of $F$?.
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1$\begingroup$ What is Tr(n,\mathbb{R})? $\endgroup$– David E SpeyerCommented Feb 20, 2012 at 18:39
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2$\begingroup$ It appears that $Tr(n,\mathbb{R})$ means (say, upper) triangular matrices. This question is a good exercise for a graduate student to work out. First, do it for 2-by-2 and maybe 3-by-3. After that, the general pattern should become clear. $\endgroup$– Deane YangCommented Feb 20, 2012 at 18:42
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