Editorially prepared introduction for this redesigned journal. This is new explanatory content, separate from Pablo's original notes.
1. Invariant directions
An eigenvector of a linear map is a nonzero vector whose direction is preserved by the transformation.
Definition. For a square matrix \(A\), a scalar \(\lambda\) is an eigenvalue if there exists \(v\ne0\) with \(Av=\lambda v\).
2. Characteristic polynomial
The equation \((A-\lambda I)v=0\) has a nonzero solution precisely when \(\det(A-\lambda I)=0\).
3. Diagonalization
If an \(n\times n\) matrix has a basis of eigenvectors, it is diagonalizable: \(A=PDP^{-1}\), where \(D\) lists the eigenvalues.
Consequently, powers become simple: \(A^k=PD^kP^{-1}\).
4. Why this matters
Eigenvalues govern growth rates in linear differential equations, stability of fixed points in dynamical systems, and resonance in physical models.