Eigenvalue and Eigenvector

An eigenvector of a linear operator is a nonzero vector that the operator merely scales, and the scaling factor is its eigenvalue, exposing the operator's action along privileged directions.

Definition

For a linear operator on a vector space, a nonzero vector is an eigenvector if the operator sends it to a scalar multiple of itself; that scalar is the corresponding eigenvalue, and it is a root of the characteristic polynomial.

Scope

This topic covers eigenvalues and eigenvectors, the characteristic and minimal polynomials, eigenspaces and algebraic versus geometric multiplicity, diagonalizability, and the spectral theorem for self-adjoint and normal operators on inner product spaces.

Core questions

• Which directions are merely scaled by a linear operator?
• How are eigenvalues found from the characteristic polynomial?
• When is an operator diagonalizable in terms of its eigenvectors?
• What special spectral structure do self-adjoint and normal operators possess?

Key theories

Characteristic polynomial

The eigenvalues of an operator are exactly the roots of its characteristic polynomial, the determinant of the operator minus a scalar times the identity, linking spectra to polynomial root-finding.

Diagonalizability criterion

An operator is diagonalizable over a field if and only if its minimal polynomial is a product of distinct linear factors over that field, equivalently when the eigenvectors span the whole space.

Spectral theorem

A self-adjoint or normal operator on a finite-dimensional inner product space has an orthonormal basis of eigenvectors and real or complex eigenvalues respectively, so it is unitarily diagonalizable.

Clinical relevance

Eigenvalues and eigenvectors describe the natural modes and stability of dynamical systems, the energy levels and observables of quantum mechanics, principal components in statistics, and the ranking vectors behind algorithms such as PageRank, making them among the most widely applied ideas in mathematics.

History

Eigenvalue problems arose in the study of quadratic forms and the principal axes of rotating bodies, with Cauchy establishing the reality of eigenvalues of symmetric matrices. Hilbert and von Neumann extended spectral theory to infinite-dimensional operators, the mathematical foundation of quantum mechanics.

Key figures

• Augustin-Louis Cauchy
• David Hilbert
• James Joseph Sylvester
• John von Neumann

Seminal works

• hoffman1971
• roman2008
• lang2002

Frequently asked questions

What is the difference between algebraic and geometric multiplicity?

Algebraic multiplicity is how many times an eigenvalue appears as a root of the characteristic polynomial; geometric multiplicity is the dimension of its eigenspace. They are equal for every eigenvalue exactly when the operator is diagonalizable.

Why does the spectral theorem matter in applications?

It guarantees that symmetric or normal operators have a full orthonormal set of eigenvectors with well-behaved eigenvalues. This underlies principal component analysis, the stability of vibrating systems, and the measurement postulates of quantum mechanics.