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I have a square matrix where the coefficients of the diagonal are 1, while the others are very small (say 10^-6). Of course the determinant of this matrix will be always one.

This is my problem:

the coefficients are functions of a variable (w, complex variable). So, the determinant of the matrix will be a function of w. I need to find the solutions, the roots w of this function (the determinant=0), but since the diagonal is made of 1 and the other coeffs are very small, this function will be constant and equal to 1.

How can I avoid this problem...extracting the diagonal...

I tried to normalize the other small coefficients using new variables but they still are very small.

Thank you for your help,

Ario