TITLE: Differences between Real Matrices and Complex matrices.
QUESTION [5 upvotes]: I am going through a course in linear algebra. Most of the time I learn that "this concept can be generalized to complex matrices without loss of generality" or "since it holds for complex matrices, it holds for real matrices also". I was curious if there are any concepts that holds for real matrices and doesn't hold for complex matrices and vice-versa.
Some trivial ones are 

Determinant and trace of a real matrix is real 
Eigen values occurs in complex conjugate pairs. 
Fundamental spaces associated with a real matrix are all real. 

thats it!!, that's all I could remember now. Any help would be appreciated.

REPLY [2 votes]: $\mathrm{GL}_n(\mathbb{R}^n)$ is disconnected, whereas $\mathrm{GL}_n(\mathbb{C}^n)$ is connected.