# Differences between Real Matrices and Complex matrices.

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.

• The second point isn't quite accurate. Non-real eigenvalues of real matrices occur in complex conjugate pairs. – Cameron Buie Nov 4 '12 at 7:55
• I think you'll find that one of the biggest differences is that $\mathbb{C}$ is algebraically closed while $\mathbb{R}$ isn't. This means that you cannot always unitarily diagonalize orthogonal and real normal matrices using purely real matrices, but unitary and normal matrices can always be unitarily diagonalized with complex matrices. – wj32 Nov 4 '12 at 7:59

$\mathrm{GL}_n(\mathbb{R}^n)$ is disconnected, whereas $\mathrm{GL}_n(\mathbb{C}^n)$ is connected.