I have a matrix equation $$ \textrm{det}(\mathbb{M}) - \textrm{tr}(\mathbb{M}) + 1 = 0 $$ where $\mathbb{M}(z)$ is a matrix function of a complex number $z$ that I want to solve for. Because I have really two equations with two unknowns (real and imaginary parts), I tried Broyden's First Method as implemented in SciPy and cannot get it to converge.
To do the math with Broyden's method requiring real-valued things, I took a guess at $z_0$ as a 2-element array, broke the equation up into real and imaginary parts, with the $0 + 0i$ being the root. Trying to figure out the value of $z$ that satisfies this.
Any pointers for using multivariate root finding schemes?