I want to calculate the complexity of an algorithm in MATLAB (not the time complexity), however, all the matrices are complex ones. I guess that the complexity of complex matrix multiplication is higher, due to the more operations required by the multiplication of complex numbers compared to real ones. Does anyone have an answer? I would like to know the same for the inversion of a complex matrix. Anyone has documentation on these or knows the big O?
In particular, I am looking for the complexity of $(A^H A)^{-1}$. A is a complex $M\times N$ matrix.
I am going to quickly type an answer here, even though almost everything has been said in the comments. As Andreas Blass already wrote: Multiplication of two complex numbers involves 4 multiplications and 2 additions of real numbers. Thus, if $f$ is the complexity of multiplying two real numbers, the complexity of multiplying two complex numbers is less than $6f$, which is still in $\mathcal{O}(f)$. You will, usually, be able to consider $f$ to be a constant, as you have a maximal size that your numbers are allowed to have (float, double, etc.). If you don't restrict number size, you have to consider the largest number in your matrix and the size of your matrix when calculating the complexity.
So, for the naive algorithm, the complexity of multiplying two matrices is going to be $\mathcal{O}(n^3)$, no matter whether it's complex or real numbers. I don't know what algorithm you use for calculating the inverse, but the same argument propably applies there: multiplication and addition within the complex numbers is going to have constant complexity and, thus, will not affect the computational complexity.