The power iteration is a classical method for computing the eigenvector associated with the largest eigenvalue of a matrix. The subspace iteration is an extension of the power iteration where the subspace spanned by n largest eigenvectors of a matrix, is determined. The natural power iteration is an exemplary instance of the subspace iteration, providing a general framework for many principal s...

We define the condition number of a nonsingular matrix on a subspace, and consider the problem of finding a subspace of given dimension that minimizes the condition number of a given matrix. We give a general solution to this problem, and show in particular that when the given dimension is less than half the dimension of the matrix, a subspace can be found on which the condition number of the m...

In the linear system Ax = b the points x are sometimes constrained to lie in a given subspace S of column space of A. Drazin inverse for any singular or nonsingular matrix, exist and is unique. In this paper, the singular consistent or inconsistent constrained linear systems are introduced and the effect of Drazin inverse in solving such systems is investigated. Constrained linear system arise ...

A unified deflating subspace approach is presented for the solution of a large class of matrix equations, including Lyapunov, Sylvester, Riccati and also some higher order polynomial matrix equations including matrix m-th roots and matrix sector functions. A numerical method for the computation of the desired deflating subspace is presented that is based on adapted versions of the periodic QZ a...

We define the condition number of a nonsingular matrix on a subspace, and consider the problem of finding a subspace of given dimension that minimizes the condition number of a given matrix. We give a general solution to this problem, and show in particular that when the given dimension is less than half the dimension of the matrix, a subspace can be found on which the condition number of the m...

In this paper we propose novel randomized subspace methods to detect anomalies in Internet Protocol networks. Given a data matrix containing information about network traffic, the proposed approaches perform a normal-plus-anomalous matrix decomposition aided by random subspace techniques and subsequently detect traffic anomalies in the anomalous subspace using a statistical test. Experimental r...

The subspace iteration algorithm, a block generalization of the classical power iteration, is known for its excellent robustness properties. Specifically, the algorithm is resilient to variations in the original matrix, and for this reason it has played an important role in applications ranging from Density Functional Theory in Electronic Structure calculations to matrix completion problems in ...

We study the stability vis a vis adversarial noise of matrix factorization algorithm for matrix completion. In particular, our results include: (I) we bound the gap between the solution matrix of the factorization method and the ground truth in terms of root mean square error; (II) we treat the matrix factorization as a subspace fitting problem and analyze the difference between the solution su...

We investigate existing and novel methods to approximate matrix functions using subspace methods. We analyze a two-sided harmonic Ritz approach and apply this to the extraction from a subspace for matrix functions. We derive all methods in various ways and provide a framework to fit in the techniques.

Subspace clustering refers to the problem of segmenting a set of data points approximately drawn from a union of multiple linear subspaces. Aiming at the subspace clustering problem, various subspace clustering algorithms have been proposed and low rank representation based subspace clustering is a very promising and efficient subspace clustering algorithm. Low rank representation method seeks ...