SymmetricEigen

SymmetricEigen is a term used for the eigenproblem of real symmetric matrices. If A is real symmetric (A = A^T), there exists an orthogonal matrix Q such that Q^T A Q = Λ, where Λ is diagonal with the eigenvalues. The columns of Q are the corresponding orthonormal eigenvectors. Equivalently, A can be written as A = Q Λ Q^T, a statement of the spectral theorem. For complex matrices, the analogous result holds for Hermitian matrices with unitary Q and real eigenvalues. Distinct eigenvalues yield mutually orthogonal eigenvectors; eigenvectors within a degenerate eigenspace can be chosen to be orthonormal.

Eigenvalues of a symmetric matrix are always real. If A is positive semidefinite, all eigenvalues are nonnegative and A has a symmetric square root. The eigenvectors provide invariant subspaces, and powers of A act diagonally in the eigenbasis.

Numerical methods for computing the symmetric eigenproblem typically first reduce A to a tridiagonal form by orthogonal similarity transforms, then apply QR-like algorithms or Jacobi iterations. The problem benefits from improved numerical stability and often smaller computational cost relative to general eigenproblems. When A is large and sparse, iterative methods like Lanczos are used.

Applications include principal component analysis, where eigenvectors of the covariance matrix give principal directions; spectral clustering and graph Laplacians; vibration analysis and modal decomposition in engineering; and quantum mechanics in certain models.