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.
Best Answer
A big one is that complex matrices always have an eigenvalue. This implies for example that all complex matrices are (unitarily) triangularizable. If you prove this property directly, then it also implies the Fundamental Theorem of Algebra.