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Prove that the set of all invertible diagonal matrices $n\times n$ over $\mathbb{C}$ is a subset path connected of $\mathbb{C}^{n^2}$.

Writing $D$ for the set of all invertible diagonal matrices $n\times n$ over $\mathbb{C}$ and given any $A,B\in D$, i must exhibit a path $f:[0,1]\rightarrow D$ such that $f$ is interely inside $D$. But this is really non-intuitive when talking about matrices. The only function that i can think of is the determinant function, but i don't know how to proceed with this idea. And also, since every $A\in D$ is invertible, its diagonal does not contain any $0$, and also i don't know how to apply this observation.

Any help would be appreciated.

Michael Hardy
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user2345678
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  • Try to modify modify every entry on the diagonal from its old value in $A$ to the new one in $B$ without passing zero, e.g. one after the other. – M. Winter Oct 14 '17 at 23:03

1 Answers1

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Let $C(\lambda)=\lambda A+(1-\lambda)B$. A polynomial $|C(\lambda)|=0$ has only finitely many roots, so there exists a path $p:[0,1]\to\Bbb C$ such that $p(0)=0$, $p(1)=1$ and $|C(p(t))|\ne 0$ for each $t\in [0,1]$. Then the path $C(p(t))$ connects $A$ and $B$ in $D$.

Alex Ravsky
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