A scalar λ for which there exists a nonzero vector v satisfying Av = λv for a linear operator or matrix A; equivalently λ is in the point spectrum of A.

Principle

Eigenvalues detect one-dimensional invariant subspaces: they are roots of the characteristic polynomial in finite dimensions and signal directions where the operator acts by scaling rather than mixing.

Demonstration

For a diagonal matrix diag(1,2,3), the eigenvalues are 1, 2 and 3, each with eigenvectors the corresponding standard basis vectors. For a 2×2 rotation matrix over R with no real eigenvectors, there are no real eigenvalues though complex ones exist.

Misapplication

Assuming every spectral value is an eigenvalue is incorrect in infinite-dimensional settings; likewise treating algebraic multiplicity as equal to geometric multiplicity without verification can mislead when Jordan blocks are present.

Consequence

Eigenvalues determine determinants and traces (product and sum of eigenvalues in finite dimensions), stability properties of dynamics (signs/moduli), and enable modal decomposition when a full eigenbasis exists.

Reversal

Contrast with singular values: singular values are nonnegative real numbers describing amplification in the Euclidean norm via A, obtained from sqrt(eigenvalues of A*A), and do not carry the same invariant-subspace interpretation as eigenvalues.

Boundary

Eigenvalues are defined relative to a chosen field and operator; over non-algebraically closed fields some eigenvalues may lie in an extension. Distinctions include algebraic vs geometric multiplicity and real vs complex eigenvalues for real operators.

Semantic Tension

Tension arises between eigenvalues and approximate/continuous spectrum: an operator may have spectral points that are not eigenvalues but still produce near-eigenbehavior; distinguishing these concepts is essential for correct analysis.

Synthesis

An eigenvalue is a scalar factor by which a linear operator stretches some nonzero vector; it is a concrete point of the point spectrum that encodes invariant directions and drives many algebraic and dynamical properties of the operator.