On an extremal property of normal matrices

We consider dynamic evaluation of algebraic functions (matrix multiplication, determinant, convolution, Fourier transform, etc.) in the model of Reif and Tate; i.e., if f(x1, ..., xn) = (y1, ..., ym) is an algebraic problem, we consider serving on-line requests of the form change input xi to value v or what is the value of output yi?. We present techniques for showing lower bounds on the worst case time complexity per operation for such problems. The first gives lower bounds in a wide range of rather powerful models (for instance history dependent algebraic computation trees over any infinite subset of a field, the integer RAM, and the generalized real RAM model of Ben-Amram and Galil). Using this technique, we show optimal Ω(n) bounds for dynamic matrix-vector product, dynamic matrix multiplication and dynamic discriminant and an Ω(√n) lower bound for dynamic polynomial multiplication (convolution), providing a good match with Reif and Tate's O(√n log n) upper bound. We also show linear lower bounds for dynamic determinant, matrix adjoint and matrix inverse and an Ω(√n) lower bound for the elementary symmetric functions. The second technique is the communication complexity technique of Miltersen, Nisan, Safra, and Wigderson which we apply to the setting of dynamic algebraic problems, obtaining similar lower bounds in the word RAM model. The third technique gives lower bounds in the weaker straight line program model. Using this technique, we show an Ω((log n)2/log log n) lower bound for dynamic discrete Fourier transform. Technical ingredients of our techniques are the incompressibility technique of Ben-Amram and Galil and the lower bound for depth-two superconcentrators of Radhakrishnan and Ta-Shma. The incompressibility technique is extended to arithmetic computation in arbitrary fields. Due to the space constraints imposed by these proceedings, in this version of the paper we only present the third technique, proving the lower bound for dynamic discrete Fourier transform and refer to the full version of the paper which is currently available as a BRICS technical report, for the rest of the proofs.

This paper discusses the upper bound analysis for the determinant of a matrix constructed using the numbers 1 to n 2. We propose four different methods to approximate the upper bound, including using the AM-GM inequality, Hadamard’s inequality, maximizing diagonal elements, and the Perron-Frobenius theorem. Our results provide important insights into the structure and properties of matrices that maximize their determinants.

Cyclical functions and permutation matrices

This paper studies the time complexity of gene expression programming based on maintaining elitist (ME-GEP). Using the theory of Markov chain and the technique of artificial fitness level, the properties of transition matrices of ME-GEP are analyzed. Based on the properties, the upper and lower bounds of the average time complexity of ME-GEP are obtained. Furthermore, the upper bound is estimated, which is determined by the parameters of ME-GEP algorithm. And the theoretical results acquired in this paper are used to analyze ME-GEP for solving function modeling and clustering problem. At last, a set of experiments are performed on these problems to illustrate the effectiveness of theoretical results. The results show that the upper bound of expected first hitting time can be used to direct the algorithm design of ME-GEP.

The aim of this study is to investigate some properties of hybrid number matrices. Firstly, we introduce hybrid numbers with some of their properties. Then we prove that any hybrid number has a 2 ? 2 complex matrix representation, and we investigate hybrid number matrices using the properties of complex matrices. Also we give answers to the following basic question ?If =I, is it true that BA =I for hybrid number matrices?? Then we define the complex adjoint matrix and the q?determinant of hybrid number matrices and give some important properties. Finally, we give an explicit formula for the inverse of a hybrid number matrix by using complex matrices.

The aim of this work is to offer necessary and sufficient conditions for the coincidence of the Drazin inverse and the DMP inverses of finite square complex matrices. This characterization is deduced from statements valid for finite potent endomorphisms and, in particular, several properties of matrices and generalized inverses are given.

It is a fundamental problem in quantum information whether a particular quantum state of a composite system is entangled. It has enormous potential in quantum error correction, quantum cryptography, and quantum teleportation applications. This problem can be transferred in the form of a mathematical conjecture in language of linear algebra. In this paper, the authors explain the important applications, convenience, efficiency of using linear algebra in math physics, and computer science. The authors give some examples of linear algebra used in various areas, including datum coordinate system finding a location, encryption and decryption algorithm, storage of images, classical mechanics, and quantum physics. The authors list the definition of the matrix, real matrix, complex matrix, diagonal matrix, identity matrix, scalar matrix, trace, rank, and determinants of matrices. The authors explored Laplace expansion, transpose, inverse, conjugate of a matrix and addition, multiplication between matrices and between a scalar and a matrix, Kronecker product and their properties of matrices, the definitions and solving method of eigenvalue and eigenvector, and the diagonalisation and its conditions of matrices. The authors introduce the applications of matrix transformations and operations in programs. The authors explain two encryption and decryption algorithms based on linear algebra and their strengths and weaknesses.

A sensitivity theory based on Fréchet derivatives is presented that has both theoretical and computational advantages. Theoretical results such as a generalization of Van Loan’s work on the matrix exponential are easily obtained: matrix functions are least sensitive at normal matrices. Computationally, the central problem is to estimate the norm of the Fréchet derivative, since this is equal to the function’s condition number. Two norm-estimation procedures are given; the first is based on a finite-difference approximation of the Fréchet derivative and costs only two extra function evaluations. The second method was developed specifically for the exponential and logarithmic functions; it is based on a trapezoidal approximation scheme suggested by the chain rule for the identity $e^X = ( e^{X/2^n } )^{2^n } $. This results in an infinite sequence of coupled Sylvester equations that, when truncated, is uniquely suited to the “scaling and squaring” procedure for $e^X $ or the “inverse scaling and squaring” procedure for log X. Both the trapezoid approximation method and the more general finite-difference approach yield excellent condition estimates for a large class of problems taken from the literature. The problems in this set illustrate that condition estimates based on the Fréchet derivative have the virtue of reliability and general applicability.

For a tree , the mean subtree order of is the average order of a subtree of . In 1984, Jamison conjectured that the mean subtree order of decreases by at least 1/3 after contracting an edge in . In this article we prove this conjecture in the special case that the contracted edge is a pendant edge. From this result, we have a new proof of the established fact that the path has the minimum mean subtree order among all trees of order . Moreover, a sharp lower bound is derived for the difference between the mean subtree orders of a tree and a proper subtree (of ), which is also used to determine the tree with second‐smallest mean subtree order among all trees of order .

A sharp lower bound is proved for the spectral gap of convex domains in the plane. Dirichlet boundary conditions are imposed on the Laplacian.

The Bridge Between Chiral Lagrangians and QCD Sum-Rules

Simultaneously normalizable matrices

Several linear and higher order methods to compute nth roots of a given real or complex matrix are presented. These include Newton-like, subspace, and Krylov type methods. As a special case, the matrix sector function and other roots of an identity matrix are computed and shown to be an efficient numerical tool for computing a block eigen-decompsition of a given matrix.

In this paper, the stability problem of sampled data Takagi–Sugeno fuzzy control systems with packet dropouts is investigated. A sampling-dependent time-varying Lyapunov function (SDTVLF) is constructed to analyze the stability problem of the system and a switched system approach is proposed to model the packet dropouts phenomenon. On this basis, by dividing the sampling input available interval and unavailable interval into several segments, the matrix functions of the SDTVLF are chosen to be continuous piecewise linear. Then, by using the proposed SDTVLF approach, computable convex conditions are obtained for the sampling input unavailable interval and the sampling input available interval in framework of dwell time. By confining the sampling input unavailable interval with an upper bound and confining the sampling input available interval with a lower bound, the SDTVLF is always decreasing in all sampling input available and unavailable interval, which can make the sampled control system tolerate a larger packet dropout rate. A numerical example is provided to show the efficiency of the proposed results.

The departure from normality of a matrix is a real scalar that is impractical to compute if the matrix is large and its eigenvalues are unknown. A simple formula is presented for computing an upper bound for departure from normality in the Frobenius norm. This new upper bound is cheaper to compute than the one derived by Henrici [Numer. Math., 4 (1962), pp. 24–40]. Moreover, the new bound is sharp for Hermitian matrices, skew-Hermitian matrices and, in general, any matrix with eigenvalues that are horizontally or vertically aligned in the complex plane. In terms of applications, the new bound can be used in computing bounds for the spectral norm of matrix functions or bounds for the sensitivity of eigenvalues to matrix perturbations.