The eigenvalues of a square matrix $A \in \mathbb{R}^{n \times n}$ cannot be analytically solved when $n \ge 5$, according to some French guy called Galois.
Directly solving the characteristic polynomial is numerically unstable, so it is not a viable algorithm.
Most stable eigenvalue algorithms are iterative, giving better and better numerical estimates to the true eigenvalues.
### Continuity of Eigenvalues
> [!bigidea]
> The eigenvalues themselves are stable under small perturbations to the matrix, so we can estimate eigenvalues by approximating the matrix itself.
The **Gerschgorin circle theorem** bounds the values of eigenvalues: any eigenvalue $\lambda$ of matrix $A \in \mathbb{R}^{n \times n}$ must lie in one of the **Gerschgorin circles/disks**: $D_{i}\equiv \left\{ z \in \mathbb{C}\,\Bigg|\, |z-a_{ii}|<\sum_{\substack{j = 1\\j \ne i}}^{n}|a_{ij}| \right\}$which is the disk centered at the $(i,i)$ entry, with radius equal to the sum of the (absolute value) of the rest of the row.
- This does not imply that every disk contains an eigenvalue: $\begin{pmatrix}1 & -1 \\ 2 & -1\end{pmatrix}$ has eigenvalues $\pm i$, but neither lies in $D_{1}=B(1,1)$.
An important corollary is that if the diagonal matrix $D=\mathrm{diag}(\lambda_{1},\dots,\lambda_{n})$ is slightly perturbed by $O(\epsilon)N$, then the new eigenvalues of $E= D + O(\epsilon)N$ can still be approximated nicely by its diagonal:
$[\text{eigenvalues of }E] \approx [\text{diagonal of }E]\approx\{ \lambda_{1},\dots,\lambda_{n} \}$
> [!proof]
> The Gerschgorin circles of $E$ have center $\lambda_{i}+O(\epsilon)n_{ii}$, and radius $O(\epsilon)\sum_{j \ne i}|n_{ij}|$, hence the new eigenvalues in those disks must be close to the original $\lambda_{i}