structural:expansion
Eigenvalues

Explicit algebraic formulas for the roots of a polynomial exist only if the degree n {\displaystyle n} is 4 or less. According to the Abel–Ruffini theorem there is no general, explicit and exact algebraic formula for the roots of a polynomial with degree 5 or more. (Generality matters because any polynomial with degree n {\displaystyle n} is the characteristic polynomial of some companion matrix of order n {\displaystyle n}.) Therefore, for matrices of order 5 or more, the eigenvalues and eigenvectors cannot be obtained by an explicit algebraic formula, and must therefore be computed by approximate numerical methods. Even the exact formula for the roots of a degree 3 polynomial is numerically impractical.