Product Of Symmetric Matrices Research Articles

Euclidean Contractivity of Neural Networks With Symmetric Weights

This paper investigates stability conditions of continuous-time Hopfield and firing-rate neural networks by leveraging contraction theory. First, we present a number of useful general algebraic results on matrix polytopes and products of symmetric matrices. Then, we give sufficient conditions for strong and weak Euclidean contractivity, i.e., contractivity with respect to the ℓ2 norm, of both models with symmetric weights and (possibly) non-smooth activation functions. Our contraction analysis leads to contraction rates which are log-optimal in almost all symmetric synaptic matrices. Finally, we use our results to propose a firing-rate neural network model to solve a quadratic optimization problem with box constraints.

Hadamard product and related inequalities in the Jordan spin algebra

Due to its simple yet elegant structure, the study of an entry-wise product of matrices, called the Hadamard product, has received extensive attention from researchers and has expanded to various disciplines, including Euclidean Jordan algebras. As an ongoing effort to extend this product to Euclidean Jordan algebras, in this article, we propose a Hadamard product in the setting of Jordan spin algebra, , under the scheme of the Peirce decomposition, and show that it preserves the diagonal structure of the elements in the algebra. It is shown that this new product corresponds to the standard Hadamard product of symmetric matrices in the case of . Lastly, we prove that this novel product satisfies an analog of the Schur product theorem as well as the inequalities of Hadamard, Oppenheim, Fiedler, etc.

Fast Bilinear Algorithms for Symmetric Tensor Contractions

Abstract In matrix-vector multiplication, matrix symmetry does not permit a straightforward reduction in computational cost. More generally, in contractions of symmetric tensors, the symmetries are not preserved in the usual algebraic form of contraction algorithms. We introduce an algorithm that reduces the bilinear complexity (number of computed elementwise products) for most types of symmetric tensor contractions. In particular, it lowers the bilinear complexity of symmetrized contractions of symmetric tensors of order s + v {s+v} and v + t {v+t} by a factor of ( s + t + v ) ! s ! ⁢ t ! ⁢ v ! {\frac{(s+t+v)!}{s!t!v!}} to leading order. The algorithm computes a symmetric tensor of bilinear products, then subtracts unwanted parts of its partial sums. Special cases of this algorithm provide improvements to the bilinear complexity of the multiplication of a symmetric matrix and a vector, the symmetrized vector outer product, and the symmetrized product of symmetric matrices. While the algorithm requires more additions for each elementwise product, the total number of operations is in some cases less than classical algorithms, for tensors of any size. We provide a round-off error analysis of the algorithm and demonstrate that the error is not too large in practice. Finally, we provide an optimized implementation for one variant of the symmetry-preserving algorithm, which achieves speedups of up to 4.58 × \times for a particular tensor contraction, relative to a classical approach that casts the problem as a matrix-matrix multiplication.

Complementarity properties of Peirce-diagonalizable linear transformations on Euclidean Jordan algebras†

Peirce-diagonalizable linear transformations on a Euclidean Jordan algebra are of the form L(x)=A·x:=∑ a ij x ij , where A=[a ij ] is a real symmetric matrix and ∑ x ij is the Peirce decomposition of an element x in the algebra with respect to a Jordan frame. Examples of such transformations include Lyapunov transformations and quadratic representations on Euclidean Jordan algebras. Schur (or Hadamard) product of symmetric matrices provides another example. Motivated by a recent generalization of the Schur product theorem, we study general and complementarity properties of such transformations.

On the equivalence of three estimators for dispersion effects in unreplicated two-level factorial designs

Box and Meyer [1986. Dispersion effects from fractional designs. Technometrics 28(1), 19–27] were the first to consider identifying both location and dispersion effects from unreplicated two-level fractional factorial designs. Since the publication of their paper a number of different procedures (both iterative and non-iterative) have been proposed for estimating the location and dispersion effects. An overview and a critical analysis of most of these procedures is given by Brenneman and Nair [2001. Methods for identifying dispersion effects in unreplicated factorial experiments: a critical analysis and proposed strategies. Technometrics 43(4), 388–405]. Under a linear structure for the dispersion effects, non-iterative estimation methods for the dispersion effects were proposed by Brenneman and Nair [2001. Methods for identifying dispersion effects in unreplicated factorial experiments: a critical analysis and proposed strategies. Technometrics 43(4), 388–405], Liao and Iyer [2000. Optimal 2 n - p fractional factorial designs for dispersion effects under a location-dispersion model. Comm. Statist. Theory Methods 29(4), 823–835] and Wiklander [1998. A comparison of two estimators of dispersion effects. Comm. Statist. Theory Methods 27(4), 905–923] (see also Wiklander and Holm [2003. Dispersion effects in unreplicated factorial designs. Appl. Stochastic. Models Bus. Ind. 19(1), 13–30]). We prove that for two-level factorial designs the proposed estimators are different representations of a single estimator. The proof uses the framework of Seely [1970a. Linear spaces and unbiased estimation. Ann. Math. Statist. 41, 1725–1734], in which quadratic estimators are expressed as inner products of symmetric matrices.

A complementary proof of an eigenvalue property in correspondence analysis

In correspondence analysis and related methods one encounters eigenvalues which lie in the [0,1] interval. This short note points out that a result on the eigenvalues of a product of symmetric matrices offers a natural proof of this property.

On the eigenvectors for a class of matrices arising from quasi-derivatives

SynopsisLet A be a product of symmetric matrices, A = RQ, with R non-singular, and let v be an eigenvector of A. For certain R and Q, a convenient formula for the expression (R−1v)tv is obtained. This expression occurs in the diagonalization of A and, in the particular case where A is associated with the quasi-derivative formulation of higher-order differential equations, the expression occurs in the asymptotic theory of solutions of the differential equation.

A basic lemma and the analysis of block and Kronecker product designs

In this paper the analysis of the class of block designs whose C matrix can be expressed in terms of the Kronecker product of some elementary matrices is considered. The analysis utilizes a basic result concerning the spectral decomposition of the Kronecker product of symmetric matrices in terms of the spectral decomposition of the component matrices involved in the Kronecker product. The property (A) of Kurkjian and Zelen (1963) is generalised and the analysis of generalised property (A) designs is given. It is proved that a design is balanced factorially if and only if it is a generalised property (A) design. A method of analysis of Kronecker product block designs whose component designs are equi-replicate and proper is also suggested.

Characterizations of products of symmetric matrices

Characterizations are obtained for matrices C of the form C = AΣ, where A, Σ are n× n matrices over the real field such that A is symmetric and C is nonnegative definite. Among others, a proof of recent generalization of Cochran's theorem is given.

Symmetric Completions and Products of Symmetric Matrices

Symmetric Completions and Products of Symmetric Matrices

Symmetric completions and products of symmetric matrices

Symmetric completions and products of symmetric matrices