A Rank 3x4 Complex Matrix Constraints

[swraman]

I am dealing with a 3x4 complex matrix M that relates a vector d to another vetor c. That is:

c = [M]*d

d is 4x1 and c is 3x1. I want to introduce a new line (constraint) into M, say d(1) = d(2). However, I would like to only apply the constraint to the real or only the imaginary parts. Is this possible?

Let me rephrase to give some perspective. M is a complex matrix that represents the frequency response of a system at a given frequency, thus M is complex as each element has magnitude and phase. Since my system has 3 outputs and 4 inputs, I want to introduce a constraint to make my matrix square (and ensure there is only one solution). I know I can set d(1) = d(2) or any relationship like that, but that locks both amplitude and phase of d(1) and d(2) together. Is there any way to set only the amplitudes together, or only the phases together?

Ideally, I'd like to set their amplitudes to a ratio of 1/10 and don't care what their phases to do whatever. This is where I get a bit murky...Is this possible?

 

[fresh_42]

You could write the entire system over the reals. This doubles your dimensions but you get a better control of individual amplitudes and phases separately. At the end you then recombine the solutions to complex numbers again.

 

[swraman]

Ah...Thats what I am looking for. Thanks. Only problem is this is to be used in a control loop, so time is of a concern.

 

[fresh_42]

Once you know how things could be simplified by your additional constraints (and watch out the $i^2$ terms!), you could go back to the then modified complex version and pretend it always has been.

Edit: For fast numerical solutions there are certainly specific algorithms which I don't know. Perhaps via writing the complex numbers as $re^{i \varphi}$ and using Taylor series for approximations. Even matrix multiplication as $(2 \times 2) \cdot (2 \times 2)$ could be reduced from $8$ to $7$ basic multiplications at the cost of additional additions.