# I'm trying to solve a system of complex equations

I'm simply trying to solve a system of equations and mathematica can't do it for all values over the region of interest. I have a 3x3 matrix that has a specific parametrization and after squaring the matrix I want to fix the eigenvalues to specific values.

matrix = (Outer[Times, {0, 0, 1}, {0, 0, 1}] + \[Epsilon]*
Outer[Times, {0, a, b}, {0, a, b}] + \[Delta]*
Outer[Times, {c, d, e}, {c, d, e}]);
matrix2[a_, b_, c_, d_, e_, \[Delta]_, \[Epsilon]_] =
matrix.Transpose[matrix];


This is how the matrix is defined. From here I assign the elements a-e with angles and then solve for the eigenvalues that correspond to specific condition.

angles = {Pi/2,Pi/4,Pi/2};
numbs = {Sin[angles[[1]]], Cos[angles[[1]]],Sin[angles[[2]]],Cos[angles[[2]]],
Sin[angles[[3]]], Cos[angles[[3]]]};
vals=Eigenvalues[matrix2[numbs[[1]],numbs[[2]],numbs[[3]],numbs[[4]]*numbs[[5]],
numbs[[4]]*numbs[[6]], \[Delta], \[Epsilon]]];


I want the ratio of the eigenvalues to be equal to a specific value and they it doesn't work except for the most specific of values, like pi/2 for all the angles, but it should for arbitrary angles between 0 - pi/2

NSolve[{vals[[3]]/vals[[1]] == ((1.12)/173.5),
vals[[2]]/vals[[1]] == ((2.9*10^-3)/173.5)},
{\[Epsilon], \[Delta]},Complexes]


Any help or advice would be greatly appreciated, I'be been stuck on this for a while and can't figure it out. If more clarification is needed, I am more than will to provide it.

Perhaps this, which converted to exact fractions trying to avoid roundoff errors:

{ToRules[N[Reduce[Simplify[{vals[[3]]/vals[[1]] == ((112/100)/(1735/10)),
vals[[2]]/vals[[1]] == ((29/10*10^-3)/(1735/10))}]]]]}


which gives you eight solutions.

• +1 note this works without converting to exact fractions, I think all that does for you is suppress a warning. ( You need parenthesis on (1735/10) btw ) – george2079 Apr 26 '16 at 14:06

In principle you can create an entirely real formulation and use FindMinimum like this:

exp = Abs[vals[[3]]/vals[[1]] - 1.12/173.5]^2 +
Abs[vals[[2]]/vals[[1]] - 2.9*10^-3/173.5]^2 /. {\[Delta] ->
dr + I di , \[Epsilon] -> er + ei I};
sol = FindMinimum[exp ,
{{dr, -5.7}, {di, 6.1}, {er, -1.4}, {ei, 1.2}}]


{1.67033*10^-7, {dr -> -5.7176, di -> 6.07862, er -> -1.32714, ei -> 1.30331}}

This is not a good solution however. It may be that there is no solution.

Edit: just to show this is sound, knowing the answer if we start close enough to a solution we get a good result

FindMinimum[exp, {{dr, .01}, {di, 0}, {er, -.1}, {ei, 0}}]


{1.2376*10^-16, {dr -> 0.00781672, di -> 6.45053*10^-7, er -> -0.0840721, ei -> -3.00245*10^-7}}