How to find a complex root from the determinate of a matrix?

Hi everyone,

Im having a trouble that can be described into two steps:

1)Im having a square complex matrix with unknown x.

2)I need to find x such that the determine of the matrix is zero.

Im trying to separate the solution into real and imag part and solve:

Function detM = detMcal(x,a,delta,L)

complexX = complex(x(1:2:end), x(2:2:end));

complexY = det(Matrix(x,a,delta,L))

detM = [real(complexY); imag(complexY)];

end

and

x = fsolve(@(x) detMcal(x,a,delta,L),[1,1]);

solution = complex(x(1:2:end), x(2:2:end))

But the program is not working. Is my code wrong or are there any better method?

Many Thanks.

Answer 1

[Dr. Seis]

Here is another example:

>>a = complex(randn(1),randn(1)); 
b = complex(randn(1),randn(1)); 
c = complex(randn(1),randn(1)); 
d = complex(randn(1),randn(1)); 
e = complex(randn(1),randn(1)); 
f = complex(randn(1),randn(1)); 
g = complex(randn(1),randn(1)); 
i = complex(randn(1),randn(1));
>> x = fsolve(@(x)det([a,b,c;d,e,f;g,x,i]),0)
Optimization terminated: first-order optimality is less than options.TolFun.

x =

      -2.1461 -     1.9256i

>> M = [a b c; d e f; g x i];
>> det(M)

ans =

   2.4603e-08 - 2.9054e-09i

>> x = (g*(b*f-c*e) + i*(a*e-b*d))/(a*f-d*c)

x =

      -2.1461 -     1.9256i

>> M = [a b c; d e f; g x i];
>> det(M)

ans =

-1.0025e-15 - 5.6272e-16i

Clearly, the version that does not use "fsolve" is giving an answer that is closer to 0. However, setting the "TolFun" to something different will allow "fsolve" to run more iterations and converge to a more optimal solution. Obviously you wouldn't actually have to define "a" through "i" and construct a matrix as I did above, you would just define the matrix with all of its complex values and then stick an "x" for the element you want to solve for. Or are you planning on reading in these values from another source?

[shane]

Hi Grant, Thanks for the explanation. I'm reading the matrix from another function. I have tried your method, and it shows that:

fsolve stopped because the problem appears to be locally singular.

<stopping criteria details >

bta =

0

And the following is my code:

bta = fsolve(@(stBeta)detMcal(stBeta,a,delta,L),0); %bta = x

where

Function detM=detMcal(stBeta,a,delta,L)

detM=det(Matrix(bta,a,delta,L)) %where Matrix is the matrix function.

I have tried use fzero and It can find the solution if it is not a complex matrix. Could you please tell me where the error is？ Thanks

[shane]

By the way, is this similar the method you did or it will give a different solution:

Beta_real = fzero('detMcal_Real',stBeta1,[],a,delta,L)

Beta_Imag = fzero('detMcal_Imag',stBeta2,[],a,delta,L)

bta = complex(Beta_real,Beta_Imag)

%%%where function detMcal_Real and detMcal_Imag find the real and imaginary part of the determinate respectively.

[Dr. Seis]

"fzero" will probably work, but is designed more for a different application. As far as what the error is, can you provide an example of what "bta", "a", "delta", "L" and/or the result of "Matrix(bta,a,delta,L)" would be?

I guess I am unsure of what you are doing when you break up the problem into real and imaginary parts since if you had a complex matrix "A", then det(A) does not necessarily equal det(real(A)) + i*det(imag(A))

Answer 2

[Dr. Seis]

You are trying to do the following, but for almost any MxM matrix and any location of x within that matrix:

Example:

M = [a b c; d e f; g x i];

We need -

0 = det(M) = g*(b*f-c*e) - x*(a*f-d*c) + i*(a*e-b*d)

Done at the command line -

>> a = complex(randn(1),randn(1)); 
>> b = complex(randn(1),randn(1)); 
>> c = complex(randn(1),randn(1)); 
>> d = complex(randn(1),randn(1)); 
>> e = complex(randn(1),randn(1)); 
>> f = complex(randn(1),randn(1)); 
>> g = complex(randn(1),randn(1)); 
>> i = complex(randn(1),randn(1));
>> x = (g*(b*f-c*e) + i*(a*e-b*d))/(a*f-d*c);
>> M = [a b c; d e f; g x i]

M =

   2.7694 + 0.1576i   3.0349 + 0.9572i  -0.0631 + 0.8003i
  -0.2050 + 0.4218i   1.4897 + 0.7922i   1.4172 + 0.6557i
  -1.2075 + 0.8491i   0.6899 + 1.2400i   1.6302 + 0.6787i

>> det(M)

ans =

   3.3931e-15 - 3.7396e-16i

[Dr. Seis]

I just want to confirm before I look into this further.

[shane]

Hi Grant,

Thanks for the reply.

I forget to mention that the matrix is dynamics. It can be 8X8 12X12 or even bigger which is depended on user input.

Is there a certain way to find equation for x?

Many Thanks.