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Square matrix
About: Square matrix is a research topic. Over the lifetime, 5000 publications have been published within this topic receiving 92428 citations.
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TL;DR: The problem of generating a matrix A with specified eigen-pair, where A is a symmetric and anti-persymmetric matrix, is presented and an existence theorem is given and proved and an expression is provided for this nearest matrix.
Abstract: The problem of generating a matrix A with specified eigen-pair, where A is a symmetric and anti-persymmetric matrix, is presented. An existence theorem is given and proved. A general expression of such a matrix is provided. We denote the set of such matrices by En. The optimal approximation problem associated with En is discussed, that is: to find the nearest matrix to a given matrix A* by A∈En. The existence and uniqueness of the optimal approximation problem is proved and the expression is provided for this nearest matrix. Copyright © 2002 John Wiley & Sons, Ltd.
31 citations
TL;DR: In this article, a new approach to the representation of nonsymmetrical optical systems by matrices is introduced, where each component of an optical system is represented by a 4 × 4 unitary matrix, and the product of those matrices yields the transfer matrix of the system.
Abstract: A new approach to the representation of nonsymmetrical optical systems by matrices is introduced. In the paraxial approximation each component of an optical system is represented by a 4 × 4 unitary matrix, and the product of those matrices yields the transfer matrix of the system. The transfer matrix that represents the propagation between two arbitrary planes through the system containing two independently rotated cylindrical lenses is decomposed into the product of three matrices. The eigenvalues of the submatrices in this factorized form determine the focal lengths of the equivalent system and the localization of the foci of the system with respect to these arbitrarily chosen planes.
31 citations
01 Jan 1993
TL;DR: A new data structure is described, the Lst@ix tree, which is a generalization to a souare matrix of McCreieht’s suffix tree ‘ior a string ordered alohabet .
Abstract: We describe a new data structure, the Lst@ix tree, which is a generalization to a souare matrix of McCreieht’s suffix tree ‘ior a string ordered alohabet . All-matrices have entries ii a totally . Based on the Lsuffix tree. we eive efficient algorithms for the static and dynamic ver&ns of the following problems that arise in many important applications in low-level image processing [19] and in visual databases [1S]: Two-Dimensional Pattern Retrieval: We have a library of texts S = {TEXT’, . . . ,TEXT’}, where TEXT’ is an ni x ni matrix, 1 < i < r. In the static version, we may preprocess the.libr&y. -Given an m x m pattern matrix PAT, we want to find all occurrences of PAT in TEXT (query), for all TEXT E S. In th e d ynamic version, we can update the librarv S bv insertine in it or deletine from it some TEXT. The query is as i; the static versi& but is intermixed with update operations. Two-Dimensional Dictionary Matching: We have a dictionary of patterns DC = {PAZ, ... , PAT,}, where PAZ is of dimension mi x mi, 1 < i < s. In the static case, we may preprocess the dictionary. liven an n x n text matrix TEXT, we want to search for all occurrences of patterns in the dictionary in the text (search step). In the dynamic version, we can update DC by inserting or deleting some matrix (pattern) from it.
31 citations
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TL;DR: The results solve the conjecture of Gunasekar et al. under the restricted isometry property, and demonstrate that the training algorithm can provide an implicit regularization for non-linear matrix factorization models.
Abstract: We study the problem of recovering a low-rank matrix $X^\star$ from linear measurements using an over-parameterized model. We parameterize the rank-$r$ matrix $X^\star$ by $UU^\top$ where $U\in \mathbb{R}^{d\times d}$ is a square matrix, whereas the number of linear measurements is much less than $d^2$. We show that with $\tilde{O}(dr^{2})$ random linear measurements, the gradient descent on the squared loss, starting from a small initialization, recovers $X^\star$ approximately in $\tilde{O}(\sqrt{r})$ iterations. The results solve the conjecture of Gunasekar et al. under the restricted isometry property, and demonstrate that the training algorithm can provide an implicit regularization for non-linear matrix factorization models.
31 citations
01 Jan 2012
TL;DR: The maximum controllability index of square matrices is defined and analyzed, and a general-ized controllable canonical form is introduced for single-input systems.
Abstract: For two types of linear time-invariant dynamical multi-agent systems un- der leader-follower framework, the problem of graph topology adjustment is addressed to improve system controllability. As important concepts and theoretical foundations, the maximum controllability index of square matrices is defined and analyzed, and a general- ized controllability canonical form is introduced for single-input systems. Based on these concepts, approaches for adjusting the leader-follower and follower-follower communica- tion architectures are presented respectively.
31 citations