In linear algebra, eigenvectors and eigenvalues characterize specific nonzero vectors that remain invariant under a matrix transformation apart from scalar scaling by their corresponding eigenvalue. This concept defines the core mechanism where a complex linear operator acts as a simple stretching operation restricted to particular subspaces (eigenspaces) spanned by its associated eigenvectors. Formally, for an $n \times n$ square matrix $A$, a nonzero vector $\mathbf{x}$ is an eigenvector if and only if it satisfies the equation $A\mathbf{x} = \lambda\mathbf{x}$, where $\lambda$ represents the real or complex scalar eigenvalue; any scalar multiple of this solution remains within the same invariant subspace.
In linear algebra, eigenvectors and eigenvalues characterize specific nonzero vectors that remain invariant under a matrix transformation apart from scalar scaling by their corresponding eigenvalue. …