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The set of invertible $k \times k$ matrices with complex entries is a connected subset of $\Bbb C^{k \times k}$.

Required Hint for this problem.

I have recently proved that the set of invertible $k \times k$ matrices with real entries is not a connected subset of $\Bbb R^{k \times k}$.

Are the two problems related??

Ref: Hoffman...Analysis in Euclidean Space Pg 76..

Thank You!!

User8976
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    It might be easier to think about the slightly stronger property of being path connected. – hardmath Jul 13 '16 at 02:26
  • Show that the Jordan form can be transformed to $I$ in three stages: First eliminate the strictly upper triangular part, then map the eigevalues straight to the unit circle and finally rotate each eigenvalue to 1. – copper.hat Jul 13 '16 at 03:10

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Hint: They are connected. If you remove an hypersurface from a real vector space, the space obtained is disconnected.

If you remove a complex hypersurface from a complex vector space, the space obtained is connected. Equivalently, you have remove a codimension 2 hypersurfaces from a n vector space

Tsemo Aristide
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