Perturbation Bounds for Matrix Eigenvalues

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Diagonals of self-adjoint operators (2005, background): “eAP = λ 1 +··· +λ n, DIAGONALS 7 the maximum being achieved on any n-dimensional projection whose range contains eigenvectors for λ 1,...,λ n. Ky Fan’s version of this result can be found on p. 22 of [Bha87]. The proof of Theorem 4.2 also requires a geometric result, asserting that if p = (p 1,...,p”
Regression on manifolds: Estimation of the exterior derivative (2011, unknown): “Weyl’s theorem [6] implies that ‖σi(Cn + λnP̃n/nh)− σi(C + λnP̃ /nh)‖(2)2 ≤ op(h).”
Spectral clustering and the high-dimensional stochastic blockmodel (2010, background): “For any symmetric matrix M , deﬁne λ ( M ) to be the eigenvalues of M and for any interval S ⊂ R , deﬁne This follows from Theorem 2.1, Weyl’s inequality [Bhatia (1987)], and the fact that the Frobenius norm is an upper bound of the spectral norm.”
Certifiably Optimal Low Rank Factor Analysis (2016, methodology): “Step 2 only relies on computing the eigenvalues of Σ − diag( u ).”