Solving symbolic equation from determinant numerically In this problem: Intersection of conics using matrix representation , a situation arises where there are two matrices (for example:)
$$Q_1 = \begin{bmatrix}1 & 0 & 0\\0 & -1 & 0\\0 & 0 & -2\end{bmatrix}$$
$$Q_2 = \begin{bmatrix}1/2 & 0 & 0\\0 & -1 & 0\\0 & 0 & -1\end{bmatrix}$$
and we must set $$det(\lambda Q_1 + \mu Q_2) = 0$$
Expanding this equation by hand is straight forward - we can group terms, rearrange, recognize the form, and solve:
$$(\lambda + \frac{\mu}{2})(-\lambda-\mu)(-2\lambda-\mu) = 0$$
(Note that the expression is not always this "nice" - i.e. when there are less zero terms in the matrices).
However, I am trying to write code to do this. Even simplifying and setting $\lambda=1$, the determinant computation still involves a symbolic variable ($\mu$). Without resorting to a big symbolic manipulation library, is there a different way to solve this procedurally?
 A: $\det(Q_1 + \mu Q_2) = \sum\limits_{k=0}^3a_k\mu^k$
$P=\sum\limits_{k=0}^3a_kX^k\in\mathbb{R}_3[X]$
You need to find the roots of $P$.
Since the polynomial is of degree $3$ or less, you can solve it using a general formula http://en.wikipedia.org/wiki/Cubic_function
A: Here is most of the process, expanding on xavierm02 said in more detail.
The determinant expression is a cubic function in $\mu$. That is
$\det(Q_1 + \mu Q_2) = \sum\limits_{k=0}^3a_k\mu^k$
If we write the matrix $$Q_1 + \mu Q_2 = \begin{bmatrix}a & b & c\\d & e & f\\g & h & i\end{bmatrix}$$ we can write $$det(Q_1 + \mu Q_2) = aei + bfg + cdh - ceg - bdi - afh$$ Now, each of $a,b,c,d,e,f,g,h,i$ have the form $\alpha + \beta \mu$, and we will refer to these $\alpha$ and $\beta$ as subscripts of the variables. For example, $aei = (a_\alpha + a_\beta\mu)(e_\alpha + e_\beta\mu)(i_\alpha + i_\beta\mu)$. Expanding this, we get $$aei = a_\alpha e_\alpha i_\alpha + a_\alpha e_\alpha i_\beta \mu +\\ a_\alpha e_\beta i_\alpha \mu + a_\alpha e_\beta i_\beta \mu^2 +\\ a_\beta e_\alpha i_\alpha \mu + a_\beta e_\alpha i_\beta \mu^2 +\\ a_\beta e_\beta i_\alpha \mu^2 + a_\beta e_\beta i_\beta \mu^3\\ = a_\alpha e_\alpha i_\alpha +\\ (a_\alpha e_\alpha i_\beta + a_\alpha e_\beta i_\alpha + a_\beta e_\alpha i_\alpha)\mu + \\ (a_\alpha e_\beta i_\beta + a_\beta e_\alpha i_\beta + a_\beta e_\beta i_\alpha) \mu^2 + \\ a_\beta e_\beta i_\beta \mu^3$$ Following this pattern, we can compute the coefficients $a_k$ 
\begin{align}
a_0 = &a_\alpha e_\alpha i_\alpha + 
b_\alpha f_\alpha g_\alpha +
c_\alpha d_\alpha h_\alpha -
c_\alpha e_\alpha g_\alpha -
b_\alpha d_\alpha i_\alpha -
a_\alpha f_\alpha h_\alpha \\
a_1 = &a_\alpha e_\alpha i_\beta + a_\alpha e_\beta i_\alpha + a_\beta e_\alpha i_\alpha +
b_\alpha f_\alpha g_\beta + b_\alpha f_\beta g_\alpha + b_\beta f_\alpha g_\alpha + \\
&c_\alpha d_\alpha h_\beta + c_\alpha d_\beta h_\alpha + c_\beta d_\alpha h_\alpha -
c_\alpha e_\alpha g_\beta + c_\alpha e_\beta g_\alpha + c_\beta e_\alpha g_\alpha - \\
&b_\alpha d_\alpha i_\beta + b_\alpha d_\beta i_\alpha + b_\beta d_\alpha i_\alpha -
a_\alpha f_\alpha h_\beta + a_\alpha f_\beta h_\alpha + a_\beta f_\alpha h_\alpha \\
a_2 = &a_\alpha e_\beta i_\beta + a_\beta e_\alpha i_\beta + a_\beta e_\beta i_\alpha +
b_\alpha f_\beta g_\beta + b_\beta f_\alpha g_\beta + b_\beta f_\beta g_\alpha + \\
&c_\alpha d_\beta h_\beta + c_\beta d_\alpha h_\beta + c_\beta d_\beta h_\alpha -
c_\alpha e_\beta g_\beta + c_\beta e_\alpha g_\beta + c_\beta e_\beta g_\alpha - \\
&b_\alpha d_\beta i_\beta + b_\beta d_\alpha i_\beta + b_\beta d_\beta i_\alpha -
a_\alpha f_\beta h_\beta + a_\beta f_\alpha h_\beta + a_\beta f_\beta h_\alpha \\
a_3 = &a_\beta e_\beta i_\beta +
b_\beta f_\beta g_\beta +
c_\beta d_\beta h_\beta -
c_\beta e_\beta g_\beta -
b_\beta d_\beta i_\beta -
a_\beta f_\beta h_\beta
\end{align}
and then use the roots equations http://en.wikipedia.org/wiki/Cubic_function#General_formula_of_roots .
