Math//eigenvalue

A scalar that captures a fundamental property of a matrix. For a square matrix A, an eigenvalue \(\lambda\) is a number such that there exists a nonzero vector *v* (the eigenvector) where A*v* = \(\lambda\)*v*. The matrix stretches the eigenvector by a factor of \(\lambda\) without changing its direction.

A scalar that captures a fundamental property of a matrix. For a square matrix A, an eigenvalue λ\lambdaλ is a number such that there exists a nonzero vector v (the eigenvector) where Av = λ\lambdaλv. The matrix stretches the eigenvector by a factor of λ\lambdaλ without changing its direction.

Eigenvalues extract the essential character of a matrix without requiring the full matrix to be computed or stored. In deep learning, the eigenvalues of the Hessian reveal the curvature of the loss landscape: positive eigenvalues indicate convex (bowl-shaped) curvature, negative eigenvalues indicate concave curvature, and a mix of both identifies saddle points.

Used across many fields: image compression (discard small eigenvalues, keep the ones that carry most information), quantum mechanics (energy levels of a system are eigenvalues of the Hamiltonian), stability analysis (eigenvalues of the Jacobian determine whether a system is stable).