Journal ArticleDOI
The Rotation of Eigenvectors by a Perturbation. III
Chandler Davis,William Kahan +1 more
TLDR
In this article, the difference between the two subspaces is characterized in terms of certain angles through which one subspace must be rotated in order most directly to reach the other, and Sharp bounds upon trigonometric functions of these angles are obtained from the gap and from bounds upon either the perturbation or a computable residual.Abstract:
When a Hermitian linear operator is slightly perturbed, by how much can its invariant subspaces change? Given some approximations to a cluster of neighboring eigenvalues and to the corresponding eigenvectors of a real symmetric matrix, and given an estimate for the gap that separates the cluster from all other eigenvalues, how much can the subspace spanned by the eigenvectors differ from the subspace spanned by our approximations? These questions are closely related; both are investigated here. The difference between the two subspaces is characterized in terms of certain angles through which one subspace must be rotated in order most directly to reach the other. These angles unify the treatment of natural geometric, operator-theoretic and error-analytic questions concerning those subspaces. Sharp bounds upon trigonometric functions of these angles are obtained from the gap and from bounds upon either the perturbation (1st question) or a computable residual (2nd question). An example is included.read more
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Journal ArticleDOI
PhaseLift: Exact and Stable Signal Recovery from Magnitude Measurements via Convex Programming
TL;DR: It is shown that in some instances, the combinatorial phase retrieval problem can be solved by convex programming techniques, and it is proved that the methodology is robust vis‐à‐vis additive noise.
Posted Content
PhaseLift: Exact and Stable Signal Recovery from Magnitude Measurements via Convex Programming
TL;DR: In this article, the authors prove that if the vectors z_i are sampled independently and uniformly at random on the unit sphere, then the signal x can be recovered exactly (up to a global phase factor) by solving a convenient semidefinite program.
Proceedings ArticleDOI
Spectral partitioning of random graphs
TL;DR: This paper shows that a simple spectral algorithm can solve all three problems above in the average case, as well as a more general problem of partitioning graphs based on edge density.
Journal ArticleDOI
Numerical methods for computing angles between linear subspaces
Ake Bjoerck,Gene H. Golub +1 more
TL;DR: Experimental results are given, which indicates that MGS gives $\theta_k$ with equal precision and fewer arithmetic operations than HT, however, HT gives principal vectors, which are orthogonal to working accuracy, which is not in general true for MGS.
Journal ArticleDOI
On the early history of the singular value decomposition
TL;DR: This paper surveys the contributions of five mathematicians who were responsible for establishing the existence of the singular value decomposition and developing its theory.
References
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Journal ArticleDOI
The rotation of eigenvectors by a perturbation
TL;DR: In this article, the behavior of the eigenvalues of a hermitian matrix under perturbation is studied and two sets of theorems of this sort are given.