I'm taking a history of algebra course and I can't seem to figure out Karl Weierstrass' Theory of Elementary Divisors. It says he began by considering two quadratic forms $\phi = \sum_{i,j=1}^n a_{ij}x_ix_j$ and $\psi = \sum_{i,j=1}^n b_{ij}x_ix_j$. And then he lets $f_0(s) = det(sA - B)$ and let $f_1(s)$ denote the greatest common divisor of the $(n-1)(n-1)$ minors of $f_0(s)$. Then $f_2(s)$ denote the greatest common divisor of the $(n-2)(n-2)$ minors of $f_0(s)$, etc. Then my book jumps to saying that $\frac{f_{i-1}(s)} {f_i(s)} = e_i(s)$ which implies that $\frac{e_{i-1}(s)} {e_i(s)}$ and that $f(s)$ differs from the product of the $e_i(s)$'s by a nonzero constant. If, then, $s_1, s_2,...,s_k$ are distinct roots of $f(s)$, it is the case that $e_i(s) = y_i\prod_{j=1}^k (s - s_j)^{m_{ij}}$ where $y_i$ is a constant and $m_{ij}\epsilon Z^{+}$. Weierstrass would later call the factors $(s - s_j)^{m_{ij}}$ the elementary divisors of $f(s)$. Weierstrass showed that if $s_j$ is a root of multiplicity m of $f_0(s) := f(s) = det(sA - B)$, then it must be a root of multiplicity greater than or equal to $m - 1$ of $f_1(s)$ and $e_i(s) = \frac{f(s)} {f_1(s)} = y_i\prod_{j=1}^k (s - s_j)^{m_{ij}}$ where $m_{ij} = 1$. It then says, put another way, $\phi$ and $\psi$ defined as above can be transformed into $\sum_{i=1}^n (x_i')^2$ and $\sum_{i=1}^n s_i(x_i')^2$, respectively.
I have spent the past 3 hours trying to research this but I can't find an article that explains it anymore in depth than what my textbook already does. I don't understand why he sets $f_0(s) = det(sA - B)$ and I really don't know where the $e_i(s) = y_i\prod_{j=1}^k (s - s_j)^{m_{ij}}$ comes from. If anyone could offer any help I would really appreciate it.