12,666 research outputs found
Low rank perturbations of quaternion matrices
Low rank perturbations of right eigenvalues of quaternion matrices are considered. For real and complex matrices it is well known that under a generic rank-k perturbation the k largest Jordan blocks of a given eigenvalue will disappear while additional smaller Jordan blocks will remain. In this paper, it is shown that the same is true for real eigenvalues of quaternion matrices, but for complex nonreal eigenvalues the situation is different: not only the largest k, but the largest 2k Jordan blocks of a given eigenvalue will disappear under generic quaternion perturbations of rank k. Special emphasis is also given to Hermitian and skew-Hermitian quaternion matrices and generic low rank perturbations that are structure-preserving
The effect of finite rank perturbations on Jordan chains of linear operators
A general result on the structure and dimension of the root subspaces of a
matrix or a linear operator under finite rank perturbations is proved: The
increase of dimension from the -th power of the kernel of the perturbed
operator to the -th power differs from the increase of dimension of the
corresponding powers of the kernels of the unperturbed operator by at most the
rank of the perturbation and this bound is sharp
Eigenvalues of rank one perturbations of unstructured matrices
Let be a fixed complex matrix and let be two vectors. The
eigenvalues of matrices form a system
of intersecting curves. The dependence of the intersections on the vectors
is studied
Performance Analysis of Spectral Clustering on Compressed, Incomplete and Inaccurate Measurements
Spectral clustering is one of the most widely used techniques for extracting
the underlying global structure of a data set. Compressed sensing and matrix
completion have emerged as prevailing methods for efficiently recovering sparse
and partially observed signals respectively. We combine the distance preserving
measurements of compressed sensing and matrix completion with the power of
robust spectral clustering. Our analysis provides rigorous bounds on how small
errors in the affinity matrix can affect the spectral coordinates and
clusterability. This work generalizes the current perturbation results of
two-class spectral clustering to incorporate multi-class clustering with k
eigenvectors. We thoroughly track how small perturbation from using compressed
sensing and matrix completion affect the affinity matrix and in succession the
spectral coordinates. These perturbation results for multi-class clustering
require an eigengap between the kth and (k+1)th eigenvalues of the affinity
matrix, which naturally occurs in data with k well-defined clusters. Our
theoretical guarantees are complemented with numerical results along with a
number of examples of the unsupervised organization and clustering of image
data
Decentralized control with input saturation: a first step toward design
This article summarizes important observations about control of decentralized systems with input saturation and provides a few examples that give insight into the structure of such systems
Rank two perturbations of matrices and operators and operator model for t-transformation of probability measures
Rank two parametric perturbations of operators and matrices are studied in
various settings. In the finite dimensional case the formula for a
characteristic polynomial is derived and the large parameter asymptotics of the
spectrum is computed. The large parameter asymptotics of a rank one
perturbation of singular values and condition number are discussed as well. In
the operator case the formula for a rank two transformation of the spectral
measure is derived and it appears to be the t-transformation of a probability
measure, studied previously in the free probability context. New transformation
of measures is studied and several examples are presented
A singular M-matrix perturbed by a nonnegative rank one matrix has positive principal minors; is it D-stable?
The positive stability and D-stability of singular M-matrices, perturbed by
(non-trivial) nonnegative rank one perturbations, is investigated. In special
cases positive stability or D-stability can be established. In full generality
this is not the case, as illustrated by a counterexample. However, matrices of
the mentioned form are shown to be P-matrices
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