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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.

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  • $\begingroup$ The second point isn't quite accurate. Non-real eigenvalues of real matrices occur in complex conjugate pairs. $\endgroup$ – Cameron Buie Nov 4 '12 at 7:55
  • $\begingroup$ 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. $\endgroup$ – wj32 Nov 4 '12 at 7:59
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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.

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

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