matlab - solving linear complex system of equations -

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suppose system of equations looks like:

a(1+2i) + b(100i) =10i; conj(a)*(11i) +b(12+ 17i)= 167;

where a , b complex numbers;

how solve a , b using matlab? (i need solve system of 10 equations.)

this math problem. have 4 equations in 4 unknowns once separate real/imag components

a*(1+2i) + b*(100i) =10i; conj(a)*(11i) +b*(12+ 17i)= 167;

is equivalent 

   real(a) - 2*imag(a)                 - 100*imag(b) =  0  2*real(a) +   imag(a) +  100*real(b)                = 10             11*imag(a) +   12*real(b)  -  17*imag(b) = 167 11*real(a)             +   17*real(b)  +  12*imag(b) = 0

then define coefficients x= [real(a) imag(a) real(b) imag(b)] in linear equations , solve them 

a= [1   -2    0  -100     2    1  100     0     0   11   12   -17    11    0   17    12];  b = [0 10 167 0]';  a\b  ans =    0.4049  14.7920  -0.0560  -0.2918

so a=0.4049+14.7929i, b = -0.0560 -0.2908i

with mean rounding errors.

of course not helpful if have 10 complex equations (20 equations in 20 unknowns). can rewrite system above 

    real(a)                 -   2*imag(a)  - 100*imag(b)  =  0             +   12*real(b)  +  11*imag(a)  -  17*imag(b)  = 167   2*real(a) +  100*real(b)  +     imag(a)                 = 10  11*real(a) +   17*real(b)                 +  12*imag(b)  = 0

and defining

d= [(1+2i)    (100i)       (11i)  (12+17i) ];   c = [10i 167]';

(from original system dx=c) 1 recognizes e*x_=f with

e = [ real(d) -imag(d)        imag(d)  real(d) ];  f =  [real(c) ; imag(c)];

except terms have signs switched due conjugation operation. can taken care off separately multiplying array e appropriate matrix of ones sign inverted terms involved in complex conjugate operation. can done follows. define matrix conj containing 1 @ locations in original pair of equations conjugate operation applied:

conj=[ 0 0        1 0];

then sign matrix is

sgn = 2* ([ imag(conj) -real(conj)             real(conj)  imag(conj)]+1 >0)-1;

and solution 

(e.*sgn)\f  ans =      0.4049    -0.0560    14.7920    -0.2918

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