Complex Rank and Real Rank of a matrix

Question

Given a matrix $X$, one can calculate its rank by finding the number of non-zero eigenvalues. How does the aforementioned notion of rank relate to complex or real rank of $X$? It would be nice to have an example where the real and complex ranks of a matrix are different.

Answer 1

For a matrix with real entries, its rank over $\Bbb C$ is the same as over $\Bbb R$, and equals to the number of nonzero complex eigenvalues (with multiplicity).

Note that this statement holds only over an algebraically closed field, so that the characteristic polynomial can be factored as $\prod_j(X-\lambda_j)$.

For example, the matrix $\pmatrix{0&-1\\1&0}$ has no real eigenvalues, though its rank is $2$.

For matrices with complex entries, the question doesn't make sense.