# From Rational Canonical Form to The Kostant Section

Suppose we have a $2\times 2$ matrix
$$M = \begin{pmatrix} x_{11} & x_{12}\\ x_{21} & x_{22} \end{pmatrix}$$
with entries in a field $F$. The characteristic polynomial of this matrix is $p(t) := {\rm det}(tI_2 – M) = t^2 – (x_{11} + x_{22})t + x_{11}x_{22} – x_{21}x_{12}$. One might ask: how can we produce a matrix with a given characteristic polynomial? This can be accomplished using the rational canonical form:
$$t^2 + at + b\mapsto \begin{pmatrix} 0 & -b\\ 1 & -a \end{pmatrix}.$$
We can calculate that the characteristic polynomial of this matrix to be $t^2 + at + b$. This map gives a bijection between quadratic monic polynomials in $F[t]$ and matrices of the above form. One way to understand this phenomenon is through algebraic groups. To explain, let's stick with $F$ having characteristic zero, though much of what we do can be done in characteristic $p$ for $p$ sufficiently large as well using very different techniques.
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