Characteristic Polynomial

$$\mathcal{X}(s) = \det(sI-A)$$ is called the characteristic polynomial of $$A \in \mathbb{R}^{n\times n}$$.
 * $$\mathcal X(s)$$ is a polynomial of degree $$n$$ with leading coefficient 1.
 * The roots of $$\mathcal X(s)$$ are the eigenvalues of $$A$$.
 * $$\mathcal X$$ has real coefficients, so eigenvalues are either real or occur in conjugate pairs.
 * There are $$n$$ eigenvalues but not necessarily $$n$$ unique eigenvalues.