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Just studying some toy examples of $2\times 2$ and $3 \times 3$ matrices, complex number multiplication already gets a bit messy.

From a numerical analysis point of view, if one were to try and build very large matrices to model physical behavior, is it much more desirable to build real matrices over complex matrices, in terms of computational cost?

What other advantages are there regarding real vs. complex matrices?

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The question is ill-posed -- you don't say what you mean by "desirable" or "superior". If a phenomenon I try to model involves only real numbers, then clearly using real matrices is the better way. On the other hand, if I try to model something in the Fourier domain, then complex numbers appear and I would try to use a matrix with complex elements, rather than split the whole thing into two matrices. Finally, if I tried to write the fastest code for matrix-matrix multiplication, I may actually try to store the real and imaginary parts separately.

So it always comes down to what exactly you're trying to do -- there is no answer so general that it is always true.

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