The energy of a graph is the sum of the singular values of its adjacency matrix. A classic inequality for singular values of a matrix sum, including its equality case, is used to study how the energy of a graph changes when edges are removed. One sharp bound and one bound that is never sharp, for the change in graph energy when the edges of a nonsingular induced subgraph are removed, are establ...

We prove an optimal estimate of the smallest singular value of a random subgaussian matrix, valid for all dimensions. For an N × n matrix A with independent and identically distributed subgaussian entries, the smallest singular value of A is at least of the order √ N − √ n − 1 with high probability. A sharp estimate on the probability is also obtained.

If the matrix of a square linear system is nonsingular but has very small singular values, then tiny perturbations of the right-hand side may cause drastic changes in the solution. We show that the probability for this to happen is very close to zero if sufficiently many singular values of the matrix are bounded away from zero.

We prove an optimal estimate on the smallest singular value of a random subgaussian matrix, valid for all fixed dimensions. For an N × n matrix A with independent and identically distributed subgaussian entries, the smallest singular value of A is at least of the order √ N − √ n − 1 with high probability. A sharp estimate on the probability is also obtained.

In this paper, a system of linear matrix equations is considered. A new necessary and sufficient condition for the consistency of the equations is derived by means of the generalized singular-value decomposition, and the explicit representation of the general solution is provided. Keywords—Matrix equation, Generalized inverse, Generalized singular-value decomposition.

We consider the algebraic Riccati equation for which the four coefficient matrices form an M -matrix K. When K is a nonsingular M -matrix or an irreducible singular M -matrix, the Riccati equation is known to have a minimal nonnegative solution and several efficient methods are available to find this solution. In this paper we are mainly interested in the case where K is a reducible singular M ...

2008 iv This dissertation applies two statistical analysis techniques for neuroimaging data. The first aim of this dissertation is to apply randomized singular value decomposition for the approximation of the top singular vectors of the singular value decomposition of a large matrix. Randomized singular value decomposition is an algorithm that approximates the top k singular vectors of a matrix...

This paper investigates the problem of state feedback H∞ control for singular systems through delta operator approach. Firstly, a bounded real lemma corresponding to a singular continuous system under the framework of the delta operator model is obtained. Then, the existence condition and explicit expression of a desirable H∞ controller for the singular delta operator system are presented. As s...

Singular values of a data in a matrix form provide insights on the structure of the data, the effective dimensionality, and the choice of hyper-parameters on higher-level data analysis tools. However, in many practical applications such as collaborative filtering and network analysis, we only get a partial observation. Under such scenarios, we consider the fundamental problem of recovering spec...

Identification of the channel matrix is of main concern in wireless MIMO (Multiple Input Multiple Output) systems. To maximize the SNR, the best way to utilize a MIMO system is to communicate on the top singular vectors of the channel matrix. Here, we present a new approach for direct blind identification of the main independent singular modes, without first estimating the channel matrix itself...