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This chapter deals with eigenvalues and eigenvectors of matrices. The material here is a sequel to Chapter 2 dealing with the solution of linear equations. Eigenvalues are very important since many engineering problems naturally lead to eigenvalue problems. When the size of a matrix is large special numerical methods are necessary for obtaining eigenvalues and eigenvectors.

In the last decade matrix polynomials have been investigated with the primary focus on adequate linearizations and good scaling techniques for computing their eigenvalues and eigenvectors. In this article we propose a new method for computing a factored Schur form of the associated companion pencil. The algorithm has a quadratic cost in the degree of the polynomial and a cubic one in the size of the coefficient matrices. Also the eigenvectors can be computed at the same cost. The algorithm is a variant of Francis’s implicitly shifted QR algorithm applied on the com- panion pencil. A preprocessing unitary equivalence is executed on the matrix polynomial to simultaneously bring the leading matrix coefficient and the constant matrix term to triangular form before forming the companion pencil. The resulting structure allows us to stably factor each matrix of the pencil as a product of k matrices of unitary-plus-rank-one form, admitting cheap and numerically reliable storage. The problem is then solved as a product core chasing eigenvalue problem. A backward error analysis is included, implying normwise backward stabil- ity after a proper scaling. Computing the eigenvectors via reordering the Schur form is discussed as well. Numerical experiments illustrate stability and efficiency of the proposed methods. ispartof: Mathematics of Computation vol:88 issue:315 pages:312-347 status: published

This letter considers the problem of estimating all the eigenvalues and eigenvectors of an irreducible matrix, corresponding to a strongly connected digraph, in the absence of knowledge on the global network topology. To this end, we propose a unified distributed strategy performed by each node in the network and relies only on the local information. The key idea is to transform the nonlinear problem of computing both the eigenvalues and eigenvectors of an irreducible matrix into a linear one. Specifically, we first transform distributively the irreducible matrix into a nonsingular irreducible matrix. Each node in the network then estimates in a distributed fashion the inverse of the nonsingular matrix by solving a set of linear equations based on a consensus-type algorithm. The eigenvalues and the corresponding eigenvectors are finally computed by exploiting the relations between the eigenvalues and eigenvectors of both the inverse and the original irreducible matrices. A numerical example is provided to demonstrate the effectiveness of the proposed distributed strategy.

We study asymptotics of generalized eigenvectors associated with Jacobi matrices. Under weak conditions on the coefficients we identify when the matrices are self-adjoint and show that they satisfy strong non-subordinacy condition., 34 pages

Linearization is a standard method often used when dealing with matrix polynomials. Recently, the concept of linearization has been extended to rational matrix functions and Fiedler-like matrix pencils for rational matrix functions have been constructed. A linearization L ( λ ) of a rational matrix function G ( λ ) does not necessarily guarantee a simple way of recovering eigenvectors of G ( λ ) from those of L ( λ ) . We show that Fiedler-like pencils of G ( λ ) allow an easy operation free recovery of eigenvectors of G ( λ ) , that is, eigenvectors of G ( λ ) are recovered from eigenvectors of Fiedler-like pencils of G ( λ ) without performing any arithmetic operations. We also consider Fiedler-like pencils of the Rosenbrock system polynomial S ( λ ) associated with an LTI system Σ in state-space form (SSF) and show that the Fiedler-like pencils allow operation free recovery of eigenvectors of S ( λ ) . The eigenvectors of S ( λ ) are the invariant zero directions of the LTI system Σ.

We present neural networks to compute the left and right eigenvectors of the real diagonalizable matrix pair with real generalized eigenvalues, corresponding to the largest or the smallest generalized eigenvalue. We establish an explicit representation for the solutions of the neural network and analyze the convergence property. We consider how to use the above neural networks for computation of the singular value problem and the generalized singular value problem. In detail, we use our neural networks to compute the left and right singular vectors of a real matrix, corresponding to the largest or the smallest singular value. The right generalized singular vector of matrix pairs, corresponding to the largest or the smallest generalized singular value, can be computed by the neural networks. Numerical examples are given to illustrate our result is reasonable.

This paper suggests one possible method to model additive type of functional equations using eigenvalues and eigenvectors of matrices with suitable numerical examples. The authors have defined a new type of Row Sum Matrix(RSM) and have discussed its eigenvalues and eigenvectors in order to model functional equations. The famous additive cauchy functional equation and Logical functional equation have also been modelled using identity matrix and Logical matrix in this study.

The eigenvectors of a fuzzy matrix correspond to steady states of a complex discrete-events system, characterized by the given transition matrix and fuzzy state vectors. The descriptions of the eigenspace (the set of all eigenvectors) for matrices in the max-min, max-ukasiewicz or max-drast fuzzy algebra have been presented in previous papers. The eigenspace of a fuzzy matrix in the max-prod algebra is investigated in this paper. First, necessary and sufficient conditions are shown under which the eigenspace restricted to increasing eigenvectors of a given matrix is non-empty, and the structure of the increasing eigenspace is described. Then, using simultaneous row and column permutations of the matrix, the complete characterization of the whole eigenspace structure of a given fuzzy matrix is shown. The details for matrices of order 3 are only presented. The method works analogously for square matrices of higher orders, with rapidly increasing complexity of the formulas.

Interval eigenvectors of interval matrices corresponding to steady states of systems with inexact data are investigated.Tolerable, strongly tolerable and weakly tolerable interval eigenvectors are described.Polynomial solutions to the recognition versions of all three types of tolerability are presented. Max-plus algebra plays a key role in the study of discrete event systems in connection with optimization problems such as scheduling or project management in which the objective function depends on the operations maximum and plus. The steady states of such systems correspond to the eigenvectors of max-plus matrices, therefore the investigation of the properties of these eigenvectors is important for applications.Matrices and vectors with interval coefficients in a max-plus algebra are studied in this paper, and polynomial algorithms are presented for the recognition problem with three types of tolerance interval eigenvector.

For generic values of q, all the eigenvectors of the transfer matrix of the U_q sl(2)-invariant open spin-1/2 XXZ chain with finite length N can be constructed using the algebraic Bethe ansatz (ABA) formalism of Sklyanin. However, when q is a root of unity (q=exp(i pi/p) with integer p>1), the Bethe equations acquire continuous solutions, and the transfer matrix develops Jordan cells. Hence, there appear eigenvectors of two new types: eigenvectors corresponding to continuous solutions (exact complete p-strings), and generalized eigenvectors. We propose general ABA constructions for these two new types of eigenvectors. We present many explicit examples, and we construct complete sets of (generalized) eigenvectors for various values of p and N., 50pp, 2 figures, v2: few typos are fixed, Nucl. Phys. B (2016)

A 2 × 2 matrix is introduced which relates the electric field at an observing site where geological distortion applies to the regional electric field, which is unaffected by the distortion. For the student of linear algebra this matrix provides a practical example with which to demonstrate the basic and important procedures of eigenvalue analysis and singular value decomposition. The significance of the results can be visualised because the eigenvectors of such a telluric distortion matrix have a clear practical meaning, as do their eigenvalues. A Mohr diagram for the distortion matrix displays when real eigenvectors exist, and tells their magnitudes and directions. The results of singular value decomposition (SVD) also have a clear practical meaning. These results too can be displayed on a Mohr diagram. Whereas real eigenvectors may or may not exist, SVD is always possible. The ratio of the two singular values of the matrix gives a condition number, useful to quantify distortion. Strong distortion causes the matrix to approach the condition known as ‘singularity’. A closely-related anisotropy number may also be useful, as it tells when a 2 × 2 matrix has a negative determinant by then having a value greater than unity.

We study the eigenscheme of a matrix which encodes information about the eigenvectors and generalized eigenvectors of a square matrix. The two main results in this paper are this decomposition encodes the numeric data of the Jordan canonical form of the matrix. We also describe how the eigenscheme can be interpreted as the zero locus of a global section of the tangent bundle on projective space. This interpretation allows one to see eigenvectors and generalized eigenvectors of matrices from an alternative viewpoint., Comment: 23 pages; v3: minor corrections, final version as appeared in Linear Algebra Appl

Eigenvalues and eigenvectors are one of the important topics over bipolar fuzzy linear algebra. In order to develop the bipolar fuzzy linear space we introduce in this article, the similarity relations, eigenvalues and eigenvectors of bipolar fuzzy matrices (BFMs). Idempotent, diagonally dominant and spectral radius of BFMs are considered here. Also, some properties and results of eigenvalues and eigenvectors for BFMs are investigated.

[EN] The relationship among eigenvalues of a given square matrix A and the rank one updated matrix A+vkq⁎, where vk is an eigenvector of A associated with the eigenvalue λk and q is an arbitrary vector, was described by Brauer in 1952. In this work we study the relations between the Jordan structures of A and A+vkq⁎. More precisely, we analyze the generalized eigenvectors of the updated matrix in terms of the generalized eigenvectors of A, as well as the Jordan chains of the updated matrix. Further, we obtain similar results when we use a generalized eigenvector of A instead of the eigenvector vk, Supported by the Spanish DGI grant MTM2013-43678-P.

Modularity Matrices for software systems can be put in block-diagonal form, where blocks are higher-level software modules, in a hierarchy of modules. But the exact module boundaries are often blurred by the uncertainty whether given matrix elements are module members or outliers. This paper provides an algorithm to determine module sizes. As a consequence the algorithm also decides which matrix elements are outliers. Matrix elements are weighted by their Affinity — an exponential function of the off-diagonality. The module size is given by the positive consecutive elements of the eigenvectors corresponding to the largest eigenvalues of this weighted symmetrized Modularity Matrix. By means of case studies, we illustrate the idea that outliers not only indicate the need for system redesign, but explicitly point out to problematic design spots. This work extends the applicability of linear algebra spectral methods to Modularity Matrices, at higher software abstraction levels than previously shown.

Numerische Mathematik, 114 (2), ISSN:0029-599X, ISSN:0945-3245

We consider an Information-Plus-Noise type matrix where the Information matrix is a spiked matrix. When some eigenvalues of the random matrix separate from the bulk, we study how the corresponding eigenvectors project onto those of the spikes. Note that, in an Appendix, we present alternative versions of the earlier results of Bai and Silverstein (Random Matrices Theory Appl 1(1):1150004, 44, 2012) (“noeigenvalue outside the support of the deterministic equivalent measure”) and Capitaine (Indiana Univ Math J 63(6):1875–1910, 2014) (“exact separation phenomenon”) where we remove some technical assumptions that were difficult to handle.

This paper concerns computing derivatives of semi-simple eigenvalues and corresponding eigenvectors of the quadratic matrix polynomial Q ( p , λ ) = λ 2 M ( p ) + λ C ( p ) + K ( p ) at p = p * . Computing derivatives of eigenvectors usually requires solving a certain singular linear system by transforming it into a nonsingular one. However, the coefficient matrix of the transformed linear system might be ill-conditioned. In this paper, we propose a new method for computing these derivatives, where the condition number of the coefficient matrix is the ratio of the maximum singular value to the minimum nonzero singular value of Q ( p * , λ ( p * ) ) , which is generally smaller than those in current literature and hence leads to higher accuracy. Numerical examples show the feasibility and efficiency of our method.

Although every nonzero vector x ∈ Cn can be an eigenvector of a nonscalar Toeplitz matrix T, this assertion is generally false if symmetry is additionally required of T. It is shown that every symmetric or skew-symmetric vector is an eigenvector of a symmetric Toeplitz (nonscalar) matrix. A problem in matrix analysis that results in the need to characterize such eigenvectors is described.

We propose to introduce the concept of exceptional points in intermediate courses on mathematics and classical mechanics by means of simple textbook examples. The first one is an ordinary second-order differential equation with constant coefficients. The second one is the well-known damped harmonic oscillator. From a strict mathematical viewpoint both are the same problem that enables one to connect the occurrence of linearly dependent exponential solutions with a defective matrix which cannot be diagonalized but can be transformed into a Jordan canonical form. Fil: Fernández, Francisco Marcelo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata. Instituto de Investigaciones Fisicoquímicas Teóricas y Aplicadas. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Instituto de Investigaciones Fisicoquímicas Teóricas y Aplicadas; Argentina

In this paper, we propose a method for obtaining eigenvectors of discrete cosine and sine transforms of types I and IV. The approach is based on constructing an initial eigenvector of one of such trigonometric transforms and a generating matrix. Multiplying powers of the matrix by the initial eigenvector, new eigenvectors are obtained. It is shown how the generated eigenvectors can be used in the fractionalization of the respective transform. Finally, we illustrate the applicability of the developed theory in the scenario of image encryption.

Before we discuss methods for computing eigenvalues, we recall a remark made in Chap. 5 A given nth-degree polynomial p(c) is the characteristic polynomial of some matrix. The companion matrix of equation ( 3.225) is one such matrix.

In the previous chapters, we have defined some numbers associated to a matrix, such as the determinant, trace, and rank. In this chapter, we focus on scalars and vectors known as eigenvalues and eigenvectors. The eigenvalues and eigenvectors have many important applications, in particular, in the study of differential equations.

It has been extensively recognized that the engineering structures are becoming increasingly precise and complex, which makes the requirements of design and analysis more and more rigorous. Therefore the uncertainty effects are indispensable during the process of product development. Besides, iterative calculations, which are usually unaffordable in calculative efforts, are unavoidable if we want to achieve the best design. Taking uncertainty effects into consideration, matrix perturbation methodpermits quick sensitivity analysis and structural dynamic re-analysis, it can also overcome the difficulties in computational costs. Owing to the situations above, matrix perturbation method has been investigated by researchers worldwide recently. However, in the existing matrix perturbation methods, correlation coefficient matrix of random structural parameters, which is barely achievable in engineering practice, has to be given or to be assumed during the computational process. This has become the bottleneck of application for matrix perturbation method. In this paper, we aim to develop an executable approach, which contributes to the application of matrix perturbation method. In the present research, the first-order perturbation of structural vibration eigenvalues and eigenvectors is derived on the basis of the matrix perturbation theory when structural parameters such as stiffness and mass have changed. Combining the first-order perturbation of structural vibration eigenvalues and eigenvectors with the probability theory, the variance of structural random eigenvalue is derived from the perturbation of stiffness matrix, the perturbation of mass matrix and the eigenvector of baseline-structure directly. Hence the Direct-Variance-Analysis (DVA) method is developed to assess the variation range of the structural random eigenvalues without correlation coefficient matrix being involved. The feasibility of the DVA method is verified with two numerical examples (one is truss-system and the other is wing structure of MA700 commercial aircraft), in which the DVA method also shows superiority in computational efficiency when compared to the Monte-Carlo method.

In material physics, surface Green’s functions are needed to analyze the electronic structures of nanoscale junctions. The algorithm for computing the functions consists of three steps. First, a matrix is generated by solving two equations. Then, the eigenvectors of the matrix are computed. Finally, another equation, which is generated by using the eigenvectors, is solved to produce a surface Green’s function. In numerical computations, a perturbation will be added into the matrix at the first step. As a result, by computing the eigenvectors of the perturbed matrix at the second step, a considerable numerical error of the function will emerge at the third step. In this paper, we modify the algorithm in order to successfully compute surface Green’s functions. We replace the first and second steps in the algorithm by an alternative step so that we can compute the eigenvectors of the matrix without computing the matrix. To show the effect of the modification, we report numerical experiments for computing the surface Green’s functions at GdAs surface using a full orbitals tight-binding model.

We discuss the perturbation analysis for eigenvalues and eigenvectors of structured homogeneous matrix polynomials with Hermitian, skew-Hermitian, H-even and H-odd structure. We construct minimal structured perturbations (structured backward errors) such that an approximate eigenvalue and eigenvector pair (finite or infinite eigenvalues) is an exact eigenvalue eigenvector pair of an appropriately perturbed structured matrix polynomial. We present various comparisons with unstructured backward errors and previous backward errors constructed for the non-homogeneous case and show that our results generalize previous results.

The eigenproblem for optimal-node matrices in max-plus algebra is shown to be solved. First, the definition of optimal-node matrix is introduced. Then for a given optimal-node matrix, it is revealed that it is easy to find all its eigenvalues and eigenvectors, which correspond to the interval time of two continuous stages and the start times of every stage of all the machines in the multi-machine interactive production process, respectively.

SUMMARY The distance of a matrix to a nearby defective matrix is an important classical problem in numerical linear algebra, as it determines how sensitive or ill-conditioned an eigenvalue decomposition of a matrix is. The concept has been discussed throughout the history of numerical linear algebra, and the problem of computing the nearest defective matrix first appeared in Wilkinsons famous book on the algebraic eigenvalue problem. In this paper, a new fast algorithm for the computation of the distance of a matrix to a nearby defective matrix is presented. The problem is formulated following Alam and Bora introduced in (2005) and reduces to finding when a parameter-dependent matrix is singular subject to a constraint. The solution is achieved by an extension of the implicit determinant method introduced by Spence and Poulton in (2005). Numerical results for several examples illustrate the performance of the algorithm. Copyright © 2013 John Wiley & Sons, Ltd.

In this article, both the eigenvectors and the eigenvalues of the q-Bernstein operators have been studied. Explicit formulae are presented for the eigenvectors, whose limit behavior is determined both in the case 0 1. Because the classical case, where q = 1, was investigated exhaustively by S. Cooper and S. Waldron back in 2000, the present article also discusses the related similarities and distinctions with the results in the classical case. Copyright © 2013 John Wiley & Sons, Ltd.

Let a ⊕ b = max ( a , b ) and a ⊗ b = a + b for a , b ∈ R ¯ = R ∪ { - ∞ } and extend these operations to matrices and vectors as in conventional algebra. We study the problems of existence and description of integer subeigenvectors (P1) and eigenvectors (P2) of a given square matrix, that is integer solutions to Ax ⩽ λ x and Ax = λ x . It is proved that P1 can be solved as easily as the corresponding question without the integrality requirement (that is in polynomial time). An algorithm is presented for finding an integer point in the max-column space of a rectangular matrix or deciding that no such vector exists. We use this algorithm to solve P2 for any matrix over R ¯ . The algorithm is shown to be pseudopolynomial for finite matrices which implies that P2 can be solved in pseudopolynomial time for any irreducible matrix. We also discuss classes of matrices for which P2 can be solved in polynomial time.

International audience; For a class of linear neutral type systems the problem of eigenvalues and eigenvectors assignment is investigated, i.e. finding the system which has the given spectrum and almost all, in some sense, eigenvectors. The result is used for the analysis of the critical number of solvability of a vector moment problem.

We consider quadratic eigenvalue problems in which the coefficient matrices, and hence the eigenvalues and eigenvectors, are functions of a real parameter. Our interest is in cases in which these functions remain differentiable when eigenvalues coincide. Many papers have been devoted to numerical methods for computing derivatives of eigenvalues and eigenvectors, but most require the eigenvalues to be well separated. The few that consider close or repeated eigenvalues place severe restrictions on the eigenvalue derivatives. We propose, analyze, and test new algorithms for computing first and higher order derivatives of eigenvalues and eigenvectors that are valid much more generally. Numerical results confirm the effectiveness of our methods for tightly clustered eigenvalues.

A fully-fledged definition for the fractional discrete Fourier transform of type IV (FDFT-IV) is presented and shown to outperform the simple definition of the FDFT-IV which is proved to be just a linear combination of the signal, its DFT-IV and their flipped versions. This definition heavily depends on the availability of orthonormal eigenvectors of the DFT-IV matrix G. An eigenanalysis is performed of a nearly tridiagonal matrix S which commutes with matrix G. An involutary unitary matrix P is defined and used for performing a similarity transformation that reduces S to a block diagonal form where the two diagonal blocks are exactly tridiagonal matrices. Moreover the elements of those two diagonal blocks are derived in order to circumvent the need for performing the two matrix multiplications involved in the similarity transformation. Orthonormal even and odd symmetric eigenvectors for S are generated - in terms of the eigenvectors of the two diagonal blocks - and proved to always be eigenvectors of G irrespective of the multiplicities of the eigenvalues of S. The relevance of the method contributed here is manifested in the case of a repeated eigenvalue of S with multiplicity 2 where a direct application of a general eigenanalysis procedure in any software package will not produce a pair of even and odd symmetric eigenvectors corresponding to this repeated eigenvalue. It should be mentioned that the almost tridiagonal matrix S which commutes with the DFT-IV matrix G being dealt with here is distinct from matrix S which commutes with the DFT matrix F dealt with in a previous paper Hanna et al. (2008) [7].

Aiming at the shortcomings of solving eigenvalues and eigenvectors of matrix by using traditional methods, this paper proposes the population migration algorithm based on speed, which can convert solving matrix eigenvalue problem to unconstrained optimization problems, and take a population migration algorithm to solve matrix eigenvalues and eigenvectors. Simulation results show that the proposed method has higher solving precision, faster convergence and stronger stability, can effectively obtain eigenvalues and eigenvectors of general matrix.

We study statistical properties of the eigenvectors of non-Hermitian random matrices, concentrating on Ginibre's complex Gaussian ensemble, in which the real and imaginary parts of each element of an N x N matrix, J, are independent random variables. Calculating ensemble averages based on the quantity $< L_\alpha | L_\beta > < R_\beta | R_\alpha >$, where $< L_\alpha |$ and $| R_\beta >$ are left and right eigenvectors of J, we show for large N that eigenvectors associated with a pair of eigenvalues are highly correlated if the two eigenvalues lie close in the complex plane. We examine consequences of these correlations that are likely to be important in physical applications., Comment: 4 pages, no figures

The partial differential equation for the three dimensional propagation of a light beam may be solved numerically by applying finite-difference techniques. We consider the matrix equation for the finite-difference, alternating direction implicit (ADI), numerical solution of the paraxial wave equation for the free-space propagation of light beams. The matrix is tridiagonal. It is also a Toeplitz matrix; Each diagonal descending from left to right is constant. Eigenvalues and eigenvectors are known for such matrices. The equation can be solved by making use of the orthogonality property of the eigenvectors.

Théorie; International audience; It is shown that all the eigenvectors of a sum of Hermitian matrices belong to the same algebraic variety. A polynomial system characterizing this variety is given and a set of nonlinear equations is derived which allows the construction of the variety. Moreover, in some specific cases, explicit expressions for the eigenvectors and eigenvalues can be obtained. Explicit solutions of selected models are also derived.

The eigenvalues and eigenvectors of a matrix have many applications in engineering and science. For example they are important in studying and solving structural problems, in the treatment of signal or image processing, in the study of quantum mechanics and in certain physical problems. It is therefore essential to analyze methodologies to obtain the eigenvectors and eigenvalues ​​of symmetric and Hermitian matrices. In this paper the authors present a methodology for obtaining the eigenvectors and eigenvalues of a symmetric or Hermitian matrix using a genetic algorithm. Unlike other methodologies, the process is centred in searching the eigenvectors and calculating the eigenvalues afterwards. In the search of the eigenvectors a genetic-based algorithm is used. Genetic algorithms are indicated when the search space is extended, unknown or with an intricate geometry. Also, the target vector space can be either real or complex, allowing in this way a wider field of application for the proposed method. The algorithm is tested comparing the results with those obtained by other methods or with the values previously known. So, seven applications are included: a real symmetric matrix corresponding to a vibrating system, a complex Hermitian matrix and an important application of the diagonalization problem (Coope matrix) corresponding to quantum mechanics examples, a physical problem in which data are analysed to reduce the number of variables, a comparison with the power method and the studies of a degenerate and an ill-conditioned matrix.

In this paper, an attempt is done to find approximate eigenvalues and the corresponding eigenvectors of three-parameter matrix eigenvalue problem by extending Rayleigh Quotient Iteration Method (RQIM), which is generally used to solve generalized eigenvalue problems of the form Ax = λBx. Convergence criteria of RQIM will be derived in terms of matrix 2-norm. We will test the computational efficiency of the Method analytically with the help of numerical examples. All calculations are done in MATLAB software.

The recent emergence of the discrete fractional Fourier transform has spurred research activity aiming at generating Hermite-Gaussian-like (HGL) orthonormal eigenvectors of the discrete Fourier transform (DFT) matrix F. By exploiting the unitarity of matrix F - resulting in the orthogonality of its eigenspaces pertaining to the distinct eigenvalues - the problem decouples into finding orthonormal eigenvectors for each eigenspace separately. A Direct Sequential Evaluation by constrained Optimization Algorithm (DSEOA) is contributed for the generation of optimal orthonormal eigenvectors for each eigenspace separately. This technique is direct in the sense that it does not require the generation of initial orthonormal eigenvectors as a prerequisite for obtaining the final optimal ones. The resulting eigenvectors are optimal in the sense of being as close as possible to samples of the Hermite-Gaussian functions. The technique is found to be numerically robust because total pivoting is allowed in performing the QR matrix decomposition step. The DSEOA is proved to be theoretically equivalent to each of the Gram-Schmidt algorithm (GSA) and the sequential orthogonal Procrustes algorithm (SOPA). However the three techniques are algorithmically quite distinct. An extensive comparative simulation study shows that the DSEOA is by far the most numerically robust technique among all sequential algorithms thus paying off for its relatively long computation time.