How to find zeros of determinant having complex parameter ?

How to find the zeros of the determinant of 4x4 matrix M which is multiplication of N complex square matrices An containing parameter.

for example not working (the matrix is made up)

An = [a   1 2+1i   n*a; 
      0   1 3+n*1i a; 
      a^2 0 1      a; 
      a   0 0      a];
M = A1....*AN
det(M) = 0

A = @(a,n) [a 1 2+1i n*a; 0 1 3+n*1i a; a^2 0 1 a; a 0 0 a];
M1 = @(a)eye(4,4);
for n = 1:10

M1 =@(a) M1(a)*A(a,n); % ?
end
M = @(a)det(M1(a));
a1 = fzero(det(M)); % fzero is not correct here (Find first few a's that satisfy det(M) = 0)

What is the most efficient way when it comes to matrix multiplication involving parameter and complex zeros?

Which function will search the complex plane for solutions ?

Answer 1

[Walter Roberson]

The solution for N = P+1 is the set of solutions for N = P together with two new values: each value of N inherits all the values for previous N and adds two more. The two new ones are the non-zero determinants of the new matrix, A(N)

So take your formula and calculate the determinant for n=N symbolically, solve that for a, and you will see two 0's and a pair of solutions in N -- a pair of roots of a quadratic.

Then just calculate "the first few" pairs over consecutive N.

[Walter Roberson]

I see that you edited your posting, but my answer stays the same.

Define your matrix symbolically. Take the symbolic determinant. solve() that in a. Remove the zeros. The result will be one or more formula in n that allow you to compute the values of a that are specific to that n.

[Ole]

Thank you. Symbolically would be better but the actual matrix has complex valued bessel functions that contain the parameter.