Abstract Linear Algebra 34 | Eigenvalues and Eigenvectors for Linear Maps

Eigenvectors and eigenvalues for a linear map are defined by a simple “scaling” condition: a nonzero vector X is an eigenvector of L if L(X) lands in the same direction as X, meaning L(X)=λX for some scalar λ. This matters because it turns a potentially complicated linear transformation into something predictable on the eigenvector directions—just multiplication by λ—so the structure of L becomes easier to analyze.

The definition works in both finite and infinite-dimensional vector spaces, without requiring any matrix representation. The only restriction is that an eigenvector cannot be the zero vector: since L(0)=0, the zero vector would satisfy the scaling condition trivially and would not carry meaningful information. For any nonzero X, the corresponding scalar λ is the eigenvalue associated with that eigenvector. Importantly, λ can be any element of the underlying field F, including 0.

A key reformulation links eigenvalues to kernels. Starting from L(X)=λX and rearranging gives (L−λI)(X)=0, where I is the identity map. That means λ is an eigenvalue exactly when the linear map L−λI has a nontrivial kernel (i.e., contains vectors other than 0). The set of all eigenvectors for a fixed eigenvalue λ is precisely the kernel of L−λI, excluding the zero vector; this kernel is called the eigenspace. Since kernels are subspaces, each eigenspace forms a subspace of V (and adding 0 keeps it a subspace).

In finite dimensions, the same eigenvalue condition can be checked using matrices. If a basis is chosen, L−λI corresponds to a matrix, and the dimension of the kernel is preserved under passing to that matrix representation. As a result, eigenvalues do not change when moving between the abstract linear map and its matrix form. Practically, this lets eigenvalues be computed by the familiar matrix method: solve the characteristic polynomial obtained from det(A−λI). The transcript emphasizes that the determinant-based approach is not just a computational trick—it matches the abstract kernel criterion.

The motivation for developing the abstract viewpoint comes from infinite-dimensional settings. A concrete example uses the vector space C^1 of real-valued functions on the real line whose derivative is continuous. On this space, differentiation is a linear map L(f)=f′. Eigenvectors arise naturally: the exponential function e^x is its own derivative, so L(e^x)=e^x, meaning e^x is an eigenvector with eigenvalue 1. This example shows that eigenvalues and eigenvectors extend beyond finite-dimensional linear algebra, even though fully characterizing them in infinite-dimensional spaces becomes more subtle—often requiring tools from functional analysis.

Overall, the core insight is that eigenvalues are exactly the scalars λ for which L−λI fails to be injective (has a nontrivial kernel), and eigenvectors are the nonzero vectors in that kernel. That single idea unifies the abstract definition, the matrix characteristic-polynomial method, and infinite-dimensional examples like differentiation on function spaces.