Nxn Matrix Research Articles

On a class of non-Hermitian matrices with positive definite Schur complements

Given a positive definite nXn matrix A and a Hermitian mXm matrix D, we characterize under which conditions there exists a strictly contractive matrix K such that the non-Hermitian block-matrix with the enties A and -AK in the first row and K^*A and D in the second has a positive definite Schur complement with respect to its submatrix A. Additionally, we show that K can be chosen such that diagonalizability of the block-matrix is guaranteed and we compute its spectrum. Moreover, we show a connection to the recently developed frame theory for Krein spaces.

The blessing of Dimensionality: Feature Selection outperforms functional connectivity-based feature transformation to classify ADHD subjects from EEG patterns of phase synchronisation.

Functional connectivity (FC) characterizes brain activity from a multivariate set of N brain signals by means of an NxN matrix A, whose elements estimate the dependence within each possible pair of signals. Such matrix can be used as a feature vector for (un)supervised subject classification. Yet if N is large, A is highly dimensional. Little is known on the effect that different strategies to reduce its dimensionality may have on its classification ability. Here, we apply different machine learning algorithms to classify 33 children (age [6-14 years]) into two groups (healthy controls and Attention Deficit Hyperactivity Disorder patients) using EEG FC patterns obtained from two phase synchronisation indices. We found that the classification is highly successful (around 95%) if the whole matrix A is taken into account, and the relevant features are selected using machine learning methods. However, if FC algorithms are applied instead to transform A into a lower dimensionality matrix, the classification rate drops to less than 80%. We conclude that, for the purpose of pattern classification, the relevant features should be selected among the elements of A by using appropriate machine learning algorithms.

Computational complexity of exterior products and multiparticle amplitudes of noninteracting fermions in entangled states

Noninteracting bosons were proposed to be used for a demonstration of quantum-computing supremacy in a boson-sampling setup. A similar demonstration with fermions would require that the fermions are initially prepared in an entangled state. I suggest that pairwise entanglement of fermions would be sufficient for this purpose. Namely, it is shown that computing multi-particle scattering amplitudes for fermions entangled pairwise in groups of four single-particle states is #P hard. In linear algebra, such amplitudes are expressed as exterior products of two-forms of rank two. In particular, a permanent of a NxN matrix may be expressed as an exterior product of N^2 two-forms of rank two in dimension 2N^2, which establishes the #P-hardness of the latter.

Complete Classification of Catalytic Branching Processes

We study a catalytic branching process (CBP) with any finite set of catalysts. This model describes a system of particles where the movement is governed by a Markov chain with arbitrary finite or countable state space and the branching may only occur at the points of catalysis. The results obtained generalize and strengthen those known in cases of CBP with a single catalyst and of branching random walk on d-dimensional integer lattice with a finite number of sources of particles generation. We propose to classify CBP with N catalysts as supercritical, critical or subcritical according to the value of the Perron root of a specified NxN matrix. Such classification agrees with the moment analysis performed here for local and total particles numbers. By introducing the criticality set C we also consider the influence of catalysts parameters on the process behavior. The proof is based on construction of auxiliary multi-type Bellman-Harris processes with the help of hitting times under taboo and on application of multidimensional renewal theorems. Keywords and phrases: catalytic branching process, classification, hitting times under taboo, moment analysis, multi-type Bellman-Harris process.

Sensor Circuit Design using Carbon Nanotube FET for Artificial Skin

This paper proposes a new sensor circuit using a 32 nm carbon nanotube FET (CNFET) technology for artificial skin. For future robotic and prosthetic applications, it is essential to develop a robust and low power artificial skin for detecting the environment through touch. Therefore, a sensor circuit for the artificial skin also has to be developed to detect the sensor signals and convert them into digital bits. The artificial skin sensor is based on a mesh of sensors consisting of a nxn matrix using CNFET, and the sensor outputs are connected to a current monitoring circuit proposed as the sensor circuit. The proposed sensor provides pressure measurements and shape information about pressure distribution.

Computing matrix inversion with optical networks

With this paper we bring about a discussion on the computing potential of complex optical networks and provide experimental demonstration that an optical fiber network can be used as an analog processor to calculate matrix inversion. A 3x3 matrix is inverted as a proof-of-concept demonstration using a fiber network containing three nodes and operating at telecomm wavelength. For an NxN matrix, the overall solving time (including setting time of the matrix elements and calculation time of inversion) scales as O(N(2)), whereas matrix inversion by most advanced computer algorithms requires ~O(N(2.37)) computational time. For well-conditioned matrices, the error of the inversion performed optically is found to be around 3%, limited by the accuracy of measurement equipment.

Purely Rickart module

A module M over a ring R is said to be purely Rickart if the right annihilator in M of each endomorphism ring of a module M is a pure submodule of M. Purely Rickart module is a proper generalization of Rickart module. Some properties of the purely Rickart module are investigated. Also, we prove that the ring nxn matrix over R is a purely Rickart ring if and only if R is a weakly n-semiherditary ring. Every n-generated projective module is purely Rickart if and only if the free R-module R (n) is a purely Rickart. Others results are provided in this paper.

Spectrum of Markov Generators on Sparse Random Graphs

We investigate the spectrum of the infinitesimal generator of the continuoustime random walk on a randomly weighted oriented graph. This is the non‐Hermitian random n × n matrix L defined by Ljk = Xjk if k ≠ j and Ljj = – Σk ≠ j Ljk, where (Xjk)j ≠ k are i.i.d. random weights. Under mild assumptions on the law of the weights, we establish convergence as n → ∞ of the empirical spectral distribution of L after centering and rescaling. In particular, our assumptions include sparse random graphs such as the oriented Erdős‐Rényi graph where each edge is present independently with probability p(n) → 0 as long as np(n) ≫ (log(n))6. The limiting distribution is characterized as an additive Gaussian deformation of the standard circular law. In free probability terms, this coincides with the Brown measure of the free sum of the circular element and a normal operator with Gaussian spectral measure. The density of the limiting distribution is analyzed using a subordination formula. Furthermore, we study the convergence of the invariant measure of L to the uniform distribution and establish estimates on the extremal eigenvalues of L.© 2014 Wiley Periodicals, Inc.

Efficient speech encryption using chaotic cat map for code-excited linear prediction based coders in packet networks

The increasing importance of multimedia applications is placing a great insist on content protection and customer privacy. Communications can be intercepted, especially over wireless links. Since encryption can effectively prevent eavesdropping, its use is widely advocated. The codec G. 729 based CS-ACELP algorithm is standardized as voice codec by ITU-T for multimedia and Voice over Internet Protocol (VoIP) applications. In this paper, we introduce a speech encryption method based chaotic cat map algorithm. Cat map extended to two-dimensional NxN matrix. It takes concepts from linear algebra and uses them to change the positions of the values of the matrix. The result after applying the Cat Map will be shuffled signals that contain the same values of the original signals. We applied our encryption scheme to the standard ITU-T G.729 standard speech coder to evaluate its performance. Simulation results show that G.729 based cat map encryption is very efficient since the encrypted speech is similar to a white noise. The perceptual evaluation of speech quality (PESQ) and enhanced modified bark spectral distortion (EMBSD) tests for speech extracted from TIMIT database confirm the efficiency of our proposed scheme.

Some Characteristics on Secondary k-normal Matrices

Concept of s-k unitary matrices is introduced. A list of about 21 conditions on an nxn matrix A, each of which is equivalent to A being s-k normal, is presented.

32-Bit NxN Matrix Multiplication: Performance Evaluation for Altera FPGA, i5 Clarkdale and Atom Pineview-D Intel General Purpose Processors

Nowadays mobile devices represent a significant portion of the market for embedded systems, and are continuously demanded in daily life. From the end-user perspective size, weight, features are the key quality criteria. These benchmarks criteria became the usual design constraints in the embedded systems design process and put a high impact on the power consumption. This paper survey and explore different low power design techniques for FPGA and processors. We compare, evaluate, and analyze, the power and energy consumption in three different designs namely, Altera FPGA Cyclone II which has a systolic array matrix multiplication implemented, i5 Clarkdale, and Atom Pineview-D Intel general purpose processors, which multiply two nxn 32-bit matrices and produce a 64-bit matrix as an output. We concluded that FPGA is a more power and energy efficient on low matrix size. However, general purpose processor performance is close to FPGA on larger matrix size as the larger cache size in general purpose processor help in reducing latency. We also concluded that the performance of FPGA can be improved in terms of latency if more systolic array processing elements are implemented in parallel to allow more concurrency.

Using gauss - Jordan elimination method with CUDA for linear circuit equation systems

Many scientific and engineering problems can use a system of linear equations. In this study, solution of Linear Circuit Equation System (LCES) for an nxn matrix using Compute Unified Device Architecture (CUDA) is described. Solution of LCES is realized on Graphics Processing Unit (GPU) instead of Central Processing Unit (CPU). CUDA is a parallel computing architecture developed by NVIDIA. Linear Circuits include resistance, impedance, capacitance, dependent - independent current sources and DC, AC voltage source. In this study, solutions of circuits that include resistance, independent current sources and DC voltage source have analyzes. Circuit analysis frequently involves solution of linear simultaneous equations that are solved Gauss-Jordan Elimination Method in this study. Gauss-Jordan Elimination is a variant of Gaussian Elimination that a method of solving a linear system equations (Ax=B). Gauss-Jordan Elimination is an algorithm for getting matrices in reduced row echelon form using elementary row operations. Gaussian Elimination has two parts. The first part (Forward Elimination) reduces a given system to triangular form. The second step uses back substitution to find the solution of the triangular echelon form system Because of elements of unknowns column matrix are dependent on each other, second step algorithm is not appropriate for parallel programming. Two parts of Gauss–Jordan Elimination are not like Gaussian Elimination's part so it is preferred. GPU implementation is more faster than solution of linear equation systems on CPU

Index Distribution of Gaussian Random Matrices

We compute analytically, for large N, the probability distribution of the number of positive eigenvalues (the index N+) of a random N x N matrix belonging to Gaussian orthogonal (beta=1), unitary (beta=2) or symplectic (beta=4) ensembles. The distribution of the fraction of positive eigenvalues c=N+/N scales, for large N, as P(c,N) approximately = exp[-betaN(2)Phi(c)] where the rate function Phi(c), symmetric around c=1/2 and universal (independent of beta), is calculated exactly. The distribution has non-Gaussian tails, but even near its peak at c=1/2 it is not strictly Gaussian due to an unusual logarithmic singularity in the rate function.

Trail to a Lyapunov equation solver

The Lyapunov matrix equation A X + X A ⊤ = B is N -stable when all eigenvalues of the real n × n matrix A have positive real part. When the real n × n matrix B is spd the solution X is spd. It is of low rank when B = C C ⊤ where C is n × r with r ≪ n . An efficient algorithm has been found for solving the low-rank equation. This algorithm is a result of over fifty years of research starting with seemingly unrelated development of alternating direction implicit (ADI) iterative solution of elliptical systems. The low rank algorithm may be applied to a full rank equation if one can approximate the right-hand side by a sum of low rank matrices. This may be attempted with the Lanczos algorithm.

On the lattice of matric-extensible radicals

A radical α in the universal class of associative rings is called matric-extensible if α (R n) = (α (R))n for any ring R, and natural number n, where R n denotes the nxn matrix ring with entries from R. We investigate matric-extensibility of the lower radical determined by a simple ring S. This enables us to find necessary and sufficient conditions for the lower radical determined by S to be an atom in the lattice of hereditary matric-extensible radicals. We also show that this lattice has atoms which are not of this form. We then describe all atoms of the lattice, and show that it is atomic.

Boxed iterations

This paper concerns iterations x k+1 = Ax k + b, where A is an n× n matrix and b an n×1 vector. After each iteration, the result is projected into a box, after which another iteration takes place. Convergence and behavior of such maps is shown. And, the iteration is considered when the matrix A and vector b are allowed to fluctuate.

Monodromy Approach to the Scaling Limits in Isomonodromy Systems

The isomonodromy deformation method is applied to the scaling limits in the linear NxN matrix equations with rational coefficients to obtain the deformation equations for the algebraic curves which describe the local behavior of the reduced versions for the relevant isomonodromy deformation equations. The approach is illustrated by the study of the algebraic curve associated to the n-large asymptotics in the sequence of the bi-orthogonal polynomials with cubic potentials.

Field theory as a matrix model

A new formulation of four dimensional quantum field theories, such as scalar field theory, is proposed as a large N limit of a special NxN matrix model. Our reduction scheme works beyond planar approximation and applies for QFT with finite number of fields. It uses quenched coordinates instead of quenched momenta of the old Eguchi-Kawai reduction known to yield correctly only the planar sector of quantum field theory. Fermions can be also included.

Ranking Portfolio Performance by a Joint Means and Variances Equality Test

Vlc propose a new procedure to rank portfolio performance. Given a set of N portfolios. we use statistical tests of dominance which produce direct mean-variance comparisons between any two portfolios in the set. These tests yield an NxN matrix of pairwise comparisons. A ranking function maps the elements of the comparison matrix into a numerical ranking. To illustrate the procedure we use a set of 133 mutual funds, including the S&PSOO index and the CRSP equal and value weighted indexes. We explore the empirical and theoretical relationships between our ranking procedure and the Treynor, Sharpe and Jensen performance measures. In general, the new procedure's ranking is relatively robust, does not allow for gaming and can be performed with small samples.

On the matricial representations of exponential operators

We present a simple method to derive matricial representations of exponentials operators exp [Gn ],where G n is a nx nmatrix. We verify that our results reproduce the simple case where G n is a 2 x 2 matrix and give new relations in the three‐dimensional case. We also discuss a matricial representation for the related Green's function when G n , for n = 2 and 3 is a real symmetric matrix.