In linear algebra, an eigenvector of a linear transformation is a non-zero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue is nothing but the factor from which the eigenvector is scaled.
Online calculators compute the eigenvalues of a square matrix by solving its characteristic equation. Now this is the equation obtained by equating to zero the characteristic polynomial.
Thus, this computational calculator first gets the characteristic equation using the polynomial calculator, then attempts to solve it analytically to obtain eigenvalues (either real or complex).
It is determined only for matrices 2×2, 3×3, and 4×4, using solutions of quadratic equation, Cubic equation and Quartic equation solution calculators respectively. Thus, we can say that it can find eigenvalues of a square matrix up to the 4th degree.
It is very unusual and rare that you have square matrix of higher degree in math problems, because, according to Abel–Ruffini theorem, a general polynomial equation of degree 5 or higher has no possible solution in radicals, thus, it can be solved only by numerical methods.