Square Roots Of Matrices Research Articles

On square roots of matrices over the max-time semiring ℝ+

The set of all nonnegative real numbers ℝ+ with respect to two operations ⊕ : (a, b) → max{a, b} and ⊙ : (a, b) → ab forms a semiring that is called a max-times semiring. A square matrix A over the semiring has a max-times square root if there is a matrix B over the semiring such that B ⊙ B = A. Whereas, a square matrix A over the semiring ℝ+ with respect to the tow usualy addition and multipliacation operation has a conventional square root if there is a matrix B over the semiring such that B × B = A. We begin this study by giving an example of an 3×3 upper triangular matrix over ℝ+ that has both conventional and max-times square roots. In this paper, we do research on necessary and sufficient conditions of matrix having max-times square roots and compare it to the conventional one. Furthermore, we determine necessary and sufficient conditions of such matrix having either one, both, or neither one. Finally, we investigate the numbers of square roots, if any, in each case under all conditions.

Square roots of H-nonnegative matrices

Roots of matrices are well-studied. The conditions for their existence are understood: The block sizes of nilpotent Jordan blocks, arranged in pairs, have to satisfy some simple algebraic property.More interesting are structured roots of structured matrices. Probably the best known example is the existence and uniqueness of positive definite square roots of a positive definite matrix. If one drops the requirement of positive definiteness of the square root, it turns out that there exists an abundance of square roots. Here a description of all canonical forms of all square roots is possible and is straightforward.H-nonnegative matrices are H-selfadjoint and are nonnegative with respect to an indefinite inner product with Gramian H. An H-nonnegative matrix B allows a decomposition in a negative definite, a nilpotent H-nonnegative, and a positive definite matrix, B=B−⊕B0⊕B+. The interesting part is B0, as only Jordan blocks of size one and two occur. Determining a square root of B reduces to determining a square root of each of B−, B0, and B+. Here we investigate for an H-nonnegative matrix: its square roots without additional structure, as well as its structured square roots that are H-nonnegative or H-selfadjoint.For these three classes of square roots of H-nonnegative matrices we show a simple criterion for their existence and describe all possible canonical forms. This is based mainly on known results but an important new part is that in all three cases we describe all possible square roots of the nilpotent H-nonnegative matrix B0 explicitly. Moreover, we show how our results can be applied to the conditional and unconditional stability of H-nonnegative square roots of H-nonnegative matrices, where the explicit description of the square roots of B0 is used.

Characterization of the critical points for the shear-stretch strain energy of a Cosserat problem

Using the embedded gradient vector field method we explicitly compute the list of critical points of the free energy for a Cosserat body model. We also formulate necessary and sufficient conditions for critical points in the abstract case of the special orthogonal group SO(n). Each critical point is then characterized using an explicit formula for the Hessian operator of a cost function defined on the orthogonal group. We also give a positive answer to an open question posed in L. Borisov, A. Fischle, P. Neff, "Optimality of the relaxed polar factors by a characterization of the set of real square roots of real symmetric matrices", ZAMM (2019), namely if all local minima of the optimization problem are global minima. We point out a few examples with physical relevance, in contrast to some theoretical (mathematical) situations that do not hold such a relevance.

Optimality of the relaxed polar factors by a characterization of the set of real square roots of real symmetric matrices

AbstractWe consider the problem to determine the optimal rotations which minimize urn:x-wiley:00442267:media:zamm201800120:zamm201800120-math-0002for a given diagonal matrix with positive entries . The objective function W is the reduced form of the Cosserat shear‐stretch energy, which, in its general form, is a contribution in any geometrically nonlinear, isotropic, and quadratic Cosserat micropolar (extended) continuum model. We characterize the critical points of the energy , determine the global minimizers and compute the global minimum. This proves the correctness of previously obtained formulae for the optimal Cosserat rotations in dimensions two and three. The key to the proof is the result that every real matrix whose square is symmetric can be written in some orthonormal basis as a block‐diagonal matrix with blocks of size at most two.

Про квадратні корені матриць над довільним полем

Про квадратні корені матриць над довільним полем

An Adaptive Square Root Unscented Kalman Filter Approach for State of Charge Estimation of Lithium-Ion Batteries

An accurate state of charge (SOC) estimation is of great importance for the battery management systems of electric vehicles. To improve the accuracy and robustness of SOC estimation, lithium-ion battery SOC is estimated using an adaptive square root unscented Kalman filter (ASRUKF) method. The square roots of the variance matrices of the SOC and noise can be calculated directly by the ASRUKF algorithm, which ensures the symmetry and nonnegative definiteness of the matrices. The process values and measurement noise covariance can be adaptively adjusted, which greatly improves the accuracy, stability, and self-adaptability of the filter. The effectiveness of the proposed method has been verified through experiments under different operating conditions. The obtained results were compared with those of extended Kalman filter (EKF) and unscented Kalman filter (UKF) , which indicates that the ASRUKF method provides better accuracy, robustness and convergence in the estimation of battery SOC for electric vehicles. The proposed method has a mean SOC estimation error of 0.5% and a maximum SOC estimation error of 0.8%. These errors are lower than those of other methods.

Dealing with ghost-free massive gravity without explicit square roots of matrices

In this paper we entertain a simple idea that the action of ghost free massive gravity (in metric formulation) depends not on the full structure of the square root of a matrix but rather on its invariants given by the elementary symmetric polynomials of the eigenvalues. In particular, we show how one can construct the quadratic action around Minkowski spacetime without ever taking the square root of the perturbed matrix. The method is however absolutely generic. And it also contains the full information on possible non-standard square roots coming from intrinsic non-uniqueness of the procedure. In passing, we mention some hard problems of those apocryphal square roots in the standard approach which might be better tackled with our method. The details of the latter are however deferred to a separate paper.

The Pell and Pell-Lucas Numbers via Square Roots of Matrices

In this paper, the Pell and Pell-Lucas numbers with specialized rational subscripts are derived from general expressions by square roots of the matrices \(M^n\) and \(N^n\). Besides, we reveal that the identities involving these numbers are produced by square roots of matrices \(M^{n/2}\) and \(N^{n/2}\). Further we show that the matrices \(M^n\) and \(N^n\) are generalized to rational powers by using the Abel's functional equation.

Distributed fusion estimation with square-root array implementation for Markovian jump linear systems with random parameter matrices and cross-correlated noises

This study presents the distributed fusion estimation of discrete-time Markovian jump linear systems with random parameter matrices and cross-correlated noises in sensor networks. The recursive linear minimum mean square error estimator is proposed based on the Gram-Schmidt orthogonalization procedure under a centralized framework. In order to avoid the loss of positive semidefiniteness and reduce dynamical range, its square-root array implementation is presented by recursively triangularizing the square roots of relevant positive semidefinite matrices. Furthermore, via the information filter form, the distributed fusion estimation with square-root array implementation is derived from the centralized fusion structure, incorporated with consensus strategy. A maneuvering target tracking simulation in a sensor network validates the proposed method.

A note on square roots of nonnegative matrices

The max-times algebra is the set R+ of nonnegative reals with operations ⊕:(a,b)→max⁡{a,b} and ⊙:(a,b)→ab. We discuss the property of matrices to be squares of max-times or conventional nonnegative matrices. We prove that there exists a matrix having a conventional nonnegative square root but no max-times square root. Also, we present a set S of cardinality three for which there is a nonnegative matrix M with spectrum S, and every such M has both conventional and max-times square roots. These results answer two questions from the recent paper by Tam and Huang.

The Jacobsthal and Jacobsthal-Lucas numbers via square roots of matrices

The Jacobsthal and Jacobsthal-Lucas numbers via square roots of matrices

Nonnegative square roots of matrices

By the square root of a (square) matrix A we mean a matrix B that satisfies B2=A. In this paper, we begin a study of the (entrywise) nonnegative square roots of nonnegative matrices, adopting mainly a graph-theoretic approach. To start with, we settle completely the question of existence and uniqueness of nonnegative square roots for 2-by-2 nonnegative matrices. By the square of a digraph H, denoted by H2, we mean the digraph with the same vertex set as H such that (i,j) is an arc if there is a vertex k such that (i,k) and (k,j) are both arcs in H. We call a digraph H a square root of a digraph G if H2=G. It is observed that a necessary condition for a nonnegative matrix to have a nonnegative square root is that its digraph has a square root, and also that a digraph G has a square root if and only if there exists a nonnegative matrix A with digraph G such that A has a nonnegative square root. We consider when or whether certain kinds of digraphs (including digraphs that are disjoint union of directed paths and circuits, permutation digraphs or a special kind of bigraphs) have square roots. We also consider when certain kinds of nonnegative matrices (including monomial matrices, rank-one matrices and nilpotent matrices with index two) have nonnegative square roots. A known characterization of loopless digraphs to have square roots, due to F. Escalantge, L. Montejano, and T. Rojano, is extended (and amended) to digraphs possibly with loops. Some natural open questions are also posed.

Real Powers of Bounded Linear Operators

This paper introduces \((I-A)^{\alpha }\) for every real number \(\alpha ,\) when A is a bounded linear operator on a Banach space \({\mathbb {X}}\) with norm of A less than 1. The principal value of \((I-A)^{\alpha }\) is defined as the limit of a series in the operator norm topology. Convergence of the series is proved using the Newton’s binomial theorem and this can be regarded as a noncommutative analogue of binomial theorem. Use this to introduce the generalized notion of spectrum of a bounded operator and finally, propose a method for finding some square roots of matrices in \(M_{n}({\mathbb {C}}),\) the set of all \(n\times n\) matrices with complex entries.

Polynomial-time algorithms for quadratic isomorphism of polynomials: The regular case

Let f=(f1,…,fm) and g=(g1,…,gm) be two sets of m≥1 nonlinear polynomials in K[x1,…,xn] (K being a field). We consider the computational problem of finding–if any–an invertible transformation on the variables mapping f to g. The corresponding equivalence problem is known as Isomorphism of Polynomials with one Secret (IP1S) and is a fundamental problem in multivariate cryptography. Amongst its applications, we can cite Graph Isomorphism (GI) which reduces to equivalence of cubic polynomials with respect to an invertible linear change of variables, according to Agrawal and Saxena. The main result is a randomized polynomial-time algorithm for solving IP1S for quadratic instances–a particular case of importance in cryptography.To this end, we show that IP1S for quadratic polynomials can be reduced to a variant of the classical module isomorphism problem in representation theory. We show that we can essentially linearize the problem by reducing quadratic-IP1S to test the orthogonal simultaneous similarity of symmetric matrices; this latter problem was shown by Chistov, Ivanyos and Karpinski (ISSAC 1997) to be equivalent to finding an invertible matrix in the linear space Kn×n of n×n matrices over K and to compute the square root in a certain representation in a matrix algebra. While computing square roots of matrices can be done efficiently using numerical methods, it seems difficult to control the bit complexity of such methods. However, we present exact and polynomial-time algorithms for computing a representation of the square root of a matrix in Kn×n, for various fields (including finite fields), as a product of two matrices. Each coefficient of these matrices lies in an extension field of K of polynomial degree. We then consider #IP1S, the counting version of IP1S for quadratic instances. In particular, we provide a (complete) characterization of the automorphism group of homogeneous quadratic polynomials. Finally, we also consider the more general Isomorphism of Polynomials (IP) problem where we allow an invertible linear transformation on the variables and on the set of polynomials. A randomized polynomial-time algorithm for solving IP when f=(x1d,…,xnd) is presented. From an algorithmic point of view, the problem boils down to factoring the determinant of a linear matrix (i.e. a matrix whose components are linear polynomials). This extends to IP a result of Kayal obtained for PolyProj.

Analysis of optical waveguide discontinuities using Denman‐Beavers iterative scheme

A new approach for solving the optical waveguide discontinuity problems is proposed, where the field-based operator techniques are employed. Instead of using Pade approximation, the square roots of the characteristic matrices are approximated by the Denman-Beavers iterative scheme with a branch-cut rotation in the complex plane. Numerical results show that the present scheme is more accurate and stable than the widely used Pade approximation.

On square roots and norms of matrices with symmetry properties

The present work concerns the algebra of semi-magic square matrices. These can be decomposed into matrices of specific rotational symmetry types, where the square of a matrix of pure type always has a particular type. We examine the converse problem of categorising the square roots of such matrices, observing that roots of either type occur, but only one type is generated by the functional calculus for matrices. Some explicit construction methods are given. Moreover, we take an observation by N.J. Higham as a motivation for determining bounds on the operator p-norms of semi-magic square matrices.

On computing complex square roots of real matrices

We present an idea for computing complex square roots of matrices using only real arithmetic.

Computing the Square Roots of a Class of Circulant Matrices

We first investigate the structures of the square roots of a class of circulant matrices and give classifications of the square roots of these circulant matrices. Then, we develop several algorithms for computing their square roots. We show that our algorithms are faster than the standard algorithm which is based on the Schur decomposition.

Structure-preserving Schur methods for computing square roots of real skew-Hamiltonian matrices

The contribution in this paper is two-folded. First, a complete characterization is given of the square roots of a real nonsingular skew-Hamiltonian matrix W. Using the known fact that every real skew-Hamiltonian matrix has infinitely many real Hamiltonian square roots, such square roots are described. Second, a structure-exploiting method is proposed for computing square roots of W, skew-Hamiltonian and Hamiltonian square roots. Compared to the standard real Schur method, which ignores the structure, this method requires significantly less arithmetic.

The computation of the square roots of circulant matrices

In the first part of this paper, we investigate the reduced forms of circulant matrices and quasi-skew circulant matrices. By using their properties we present two efficient algorithms to compute the square roots of circulant matrices and quasi-skew circulant matrices, respectively. Those methods are faster than the traditional algorithm which is based on the Schur decomposition. In the second part, we further consider circulant H-matrices with positive diagonal entries and develop two algorithms for computing their principal square roots. Those two algorithms have the common advantage that is they only need matrix–matrix multiplications in their iterative sequences, an operation which can be done very efficiently on modern high performance computers.