A Relative Perturbation Bound for Positive De nite Matrices

We give a sharp estimate for the eigenvectors of a positive deenite Hermitian matrix under a oating-point perturbation. The proof is elementary. Recently there have been a number of papers on eigenvector perturbation bounds that involve a perturbation of the matrix which is small in some relative sense, including the typical rounding errors in matrix elements ((1], 2], 10], 8], 4], 5]). The proofs are mostly complicated and all of them involve the notion of`the relative gap' between the eigenvalues. i.e. a relative distance of the unperturbed eigenvalue to the rest of the spectrum. Several such relative gaps are in use. Anyway, in any such estimate it is only the nearest eigenvalue that matters, one does not care for distant eigenvalues and their innuence. Our bounds control primarily the angles between the perturbed and the unperturbed eigenvectors (standard bounds with relative gaps may be derived from them any time). In particular, the distant eigenvalues naturally damp out the perturbation of the corresponding components of the eigenvector. The bounds are asymptotically sharp i.e. for small perturbations they reach the rst term of the perturbation theory. Our proof is simple (of all works cited above 4], 5] are the closest to ours)-the only technical tool is the square root of a positive deenite matrix. The simplicity of our proof may make it useful in a classroom.