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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 notion of symmetric stiffness matrices has been introduced in this article, where it is shown that a stiffness matrix should be symmetric and either positive semi-definite or positive definable.
Abstract: Discussed in this paper is the Cartesian stiffness matrix, which recently has received special attention within the robotics research community. Stiffness is a fundamental concept in mechanics; its representation in mechanical systems whose potential energy is describable by a finite set of generalized coordinates takes the form of a square matrix that is known to be, moreover, symmetric and positive-definite or, at least, semi-definite. We attempt to elucidate in this paper the notion of “asymmetric stiffness matrices”. In doing so, we show that to qualify for a stiffness matrix, the matrix should be symmetric and either positive semi-definite or positive-definite. We derive the conditions under which a matrix mapping small-amplitude displacement screws into elastic wrenches fails to be symmetric. From the discussion, it should be apparent that the asymmetric matrix thus derived cannot be, properly speaking, a stiffness matrix. The concept is illustrated with an example.
84 citations
TL;DR: In this paper, the group inverse of a matrix is used to define the #-order on square matrices of index 1, which is similar to the ∗-order of Drazin [2] and the minus order of Hartwig [6, 10] and Nambooripad [17].
84 citations
30 Apr 2008
TL;DR: In this article, Chen et al. showed that the higher rank numerical range of a normal matrix is a convex polygon determined by the eigenvalues and that the set can be obtained as the intersection of closed half planes (of complex numbers).
Abstract: CHI-KWONG LI AND NUNG-SING SZEAbstract. Results on matrix canonical forms are used to give a completedescription of the higher rank numerical range of matrices arising from thestudy of quantum error correction. It is shown that the set can be obtainedas the intersection of closed half planes (of complex numbers). As a result,it is always a convex set in C. Moreover, the higher rank numerical range ofa normal matrix is a convex polygon determined by the eigenvalues. Thesetwo consequences confirm the conjectures of Choi et al. on the subject. Inaddition, the results are used to derive a formula for the optimal upper boundfor the dimension of a totally isotropic subspace of a square matrix, and verifythe solvability of certain matrix equations.
83 citations
TL;DR: The Perron-Frobenius Theorem says that if A is a nonnegative square matrix some power of which is positive, then there exists an x0 such that Anx/‖Anx‖ converges to xn for all x > 0.
Abstract: The Perron-Frobenius Theorem says that if A is a nonnegative square matrix some power of which is positive, then there exists an x0 such that Anx/‖Anx‖ converges to xn for all x > 0. There are many classical proofs of this theorem, all depending on a connection between positively of a matrix and properties of its eigenvalues. A more modern proof, due to Garrett Birkhoff, is based on the observation that every linear transformation with a positive matrix may be viewed as a contraction mapping on the nonnegative orthant. This observation turns the Perron-Frobenius theorem into a special ease of the Banach contraction mapping theorem. Furthermore, it applies equally to linear transformations which are positive in a much more general sense. The metric which Birkhoff used to show that positive linear transformations correspond to contraction mappings is known as Hilbert's projective metric. The definition of this metric is rather complicated. It is therefore natural to try to define another, less complicated m...
83 citations
TL;DR: It is suggested that the spectral properties near zero virtuality of three-dimensional QCD follow from a Hermitian random matrix model, which is a family of random matrix models for both even and odd number of fermions.
Abstract: We suggest that the spectral properties near zero virtuality of three-dimensional QCD follow from a Hermitian random matrix model. The exact spectral density is derived for this family of random matrix models for both even and odd number of fermions. New sum rules for the inverse powers of the eigenvalues of the Dirac operator are obtained. The issue of anomalies in random matrix theories is discussed.
82 citations