Eigh (Symmetric)
Shippingmatrix_eighEigen-decomposition of a symmetric/Hermitian matrix: real eigenvalues and orthonormal vectors
Signature
Inputs
aSignal|Arrayrequired— A symmetric (Hermitian) 2-D matrix.
Outputs
eigenvaluesVector— The real eigenvalues $\lambda_i$ (ascending).eigenvectorsMatrix— The orthonormal eigenvectors as columns.
Description
Eigh (Symmetric) computes the eigen-decomposition of a symmetric (Hermitian) matrix via the specialized MKL LAPACK symmetric solver. Because the input is assumed symmetric, the eigenvalues are guaranteed real and the eigenvectors form an orthonormal basis — so this node is faster and more accurate than the general matrix_eig whenever symmetry holds (covariance matrices, Gram matrices, Hessians, graph Laplacians).
Outputs are the real eigenvalues vector and the eigenvectors matrix (columns are the vectors). Both are dimensionless. Only the symmetric part is used, so a slightly non-symmetric input is treated as its symmetrization rather than rejected.
Mathematics
Examples
Diagonalizing a symmetric matrix
For the symmetric , the eigenvalues output sums to the trace , and the eigenvectors columns are mutually orthonormal — a valid orthogonal diagonalization .
Applications
- Principal component analysis on a covariance matrix (eigenvectors = principal axes).
- Modal analysis and vibration modes of symmetric stiffness/mass systems.
- Spectral graph methods on a symmetric Laplacian.
- Checking positive-definiteness by inspecting the sign of the eigenvalues.
Neat
Exploiting symmetry guarantees real eigenvalues and orthonormal vectors, sidestepping the complex machinery that matrix_eig needs for general matrices.
The eigenvalues sum to the trace, giving a free consistency check against matrix_trace.
Known issues
Only the symmetric part of the input is used; feeding a genuinely non-symmetric matrix silently analyzes its symmetrization — use matrix_eig for general matrices.
Individual eigenvector signs are not unique and may flip between runs while remaining a valid basis.