# Real matrix is diagonalizable on $\mathbb{R}$ if it is diagonalizable on $\mathbb{C}$ and all eigenvalues are real.

Suppose we are given a real matrix $$A$$. We know that we may regard it as a complex matrix (with real entries). Every complex matrix is similar to a Jordan form where all eigenvalues are placed on diagonal. Suppose the Jordan form $$\Lambda$$ of $$A$$ is real and diagonal. This implies $$A$$ is diagonalizable on $$\mathbb{C}$$: $$PAP^{-1}=\Lambda$$. The problem I 'm having is that $$P$$ may be complex so it does not directly imply $$A$$ is similar to diagonal $$\Lambda$$ even though $$\Lambda$$ is real--when we are now working with real instead of just complex field. If all diagonal entries on $$\Lambda$$ are distinct reals, this would imply diagonalizable over the real trivially. But what if there are eigenvalues with larger than 1 multiplicity?

If $$A$$ is diagonalizable over the complex numbers and its eigenvalues are real, its minimal polynomial is $$(X-c_1)...(X-c_p)$$ where $$c_1,...,c_p$$ are the eigenvalues. Apply