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Well, I am not getting any hint how to show $GL_n(\mathbb{C})$ is path connected. So far I have thought that let $A$ be any invertible complex matrix and $I$ be the idenity matrix, I was trying to show a path from $A$ to $I$ then define $f(t)=At+(1-t)I$ for $t\in[0,1]$ which is possible continous except where the $\operatorname{det}{f(t)}=0$ i.e. which has $n$ roots and I can choose a path in $\mathbb{C}\setminus\{t_1,\dots,t_n\}$ where $t_1,\dots,t_n$ are roots of $\operatorname{det}{f(t)}=0$, is my thinking was correct? Could anyone tell me the solution?

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This path can fail in liying on $GL_n(\mathbb{C})$. For example, if you take $A=-I$, at the middle point $t=1/2$ you will get $f(t)=0$ –  matgaio May 1 '12 at 19:45
    
+1 Looks good to me. @matgaio, the idea seems to be that you *go around that troublesome $t=1/2$* - there is room for that, when you let $t$ be a complex number! It might have been clearer to define $f(t)$ for all complex $t$, exclude the finite set of points, and then select a path from $0$ to $1$. –  Jyrki Lahtonen May 1 '12 at 19:47
    
Uhm, ok. I was understanding he was trying to go straight from $A$ to $I$. –  matgaio May 1 '12 at 19:50
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This thread is slightly relevant. –  Antonio Vargas May 1 '12 at 20:02
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+1 for your intelligence. –  Dutta Dec 16 '13 at 14:06
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4 Answers 4

up vote 10 down vote accepted
  • If $P$ is a polynomial of degree $n$, the set $\{\lambda,P(\lambda)\neq 0\}$ is path connected (because its complement is finite, so you can pick a polygonal path).
  • Let $P(t):=\det(A+t(I-A))$. We have that $P(0)=\det A\neq 0$, and $P(1)=\det I=1\neq 0$, so we can find a path $\gamma\colon[0,1]\to\mathbb C$ such that $\gamma(0)=0$, $\gamma(1)=1$, and $P(\gamma(t))\neq 0$ for all $t$. Finally, put $\Gamma(t):=A+\gamma(t)(I-A)$.
  • If $B_1$ and $B_2$ are two invertible matrices, consider $\gamma(t):=B_2\cdot\gamma(t)$, where we chose $\gamma$ for $A:=B_2^{-1}B_1$.
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Since any matrix $A\in GL_n(\mathbb C)$ has only finitely many eigenvalues, and 0 isn't one of them, there is a point $z\in S^1$ such that the line through the origin containing $z$ doesn't intersect any of the eigenvalues of $A$. Now, consider the path $f(t)=At+z(1-t)I$. This has determinant 0 iff $z(t-1)$ is an eigenvalue of $At$, which happens iff $z(1-1/t)$ is an eigenvalue of $A$ (this doesn't work when $t=0$, but then it is clear that the determinant is non-zero). By construction, it isn't for any $t\in[0,1]$ so this defines a path form $A$ to $zI$. now there is a path not passing through 0 from $z$ to 1, and this gives rise to a path from $zI$ to $I$, and so concatenating the two paths, we get a path from $A$ to $I$, showing that $GL_n(\mathbb C)$ is path connected.

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Use $\Gamma(t) = e^{t \log A + (1-t) \log B}$. This is well defined since $A,B$ are invertible. $\Gamma(t)$ is clearly invertible for all $t$, $\Gamma(0) = B$, $\Gamma(1) = A$.

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This idea is not mine, I just can't remember where I saw it, it was in a control theory context. –  copper.hat May 1 '12 at 20:20
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Following idea shows connectedness of $Gl(n,\mathbb{C})$ ..

As $A\in Gl(n,\mathbb{C})$ you do have an upper triangular matrix which is similar to $A$.

See subgroup of Invertible, Upper triangular matrices as

$$\underbrace{\mathbb{C}^*\times\mathbb{C}^*\times \cdots\times\mathbb{C}^*}_{n- times}\times \underbrace{\mathbb{C}\times\mathbb{C}\times \mathbb{C}\times \cdots \times\mathbb{C}}_{\dfrac{n(n-1)}{2}times}$$

As $\mathbb{C}^*$ is connected and $\mathbb{C}$ is connected so is the above product.

See that conjugation is continuous so preserves connectedness.

By which i mean that $\{BUB^{-1} : U - \text{Upper Triangular}\}$ is connected.

See that $Gl(n,\mathbb{C})$ being union of connected sets is also connected. (Really?)

Qn : What element do you see common in all those conjugates?

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