Inverting a complex matrix

We analyze a complex matrix inversion algorithm proposed by Frobenius, which we call the Frobenius inversion. We show that the Frobenius inversion uses the least number of real matrix multiplications and inversions among all complex matrix inversion algorithms. We also analyze numerical properties of the Frobenius inversion. We prove that the Frobenius inversion runs faster than the widely used method based on LU decomposition if and only if the ratio of the running time of the real matrix inversion to that of the real matrix multiplication is greater than $5/4$. We corroborate this theoretical result by numerical experiments. Moreover, we apply the Frobenius inversion to matrix sign function, Sylvester equation, and polar decomposition. In each of these examples, the Frobenius inversion is more efficient than inversion via LU-decomposition.