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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 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.

In this study, some algebraic characterizations of the coefficient matrix A of the planar three-index transportation problem are derived and the equivalent formulation of this problem is obtained using the Kronecker product. It is shown that eigenvectors of the matrix G(+)G are characterized in terms of eigenvectors of the matrix A(+)A, where G(+) is the Moore-Penrose inverse of the coefficient matrix G of the equivalent problem.

By using exponential functions and the theory of ordinary differential equations, the classical theorem about the existence of a Jordan canonical form of a square matrix is proved.

(2007). The Probability that a Matrix of Integers Is Diagonalizable. The American Mathematical Monthly: Vol. 114, No. 6, pp. 491-499.

A nonsymmetric N × N matrix with elements as certain simple functions of N distinct real or complex numbers r 1, r 2, …, rN is presented. The matrix is special due to its eigenvalues − the consecu ...

In the hydrodynamic stability analysis of fluid flow, an eigenproblem with a large asymmetric matrix can be produced by the linearized Navier-Stokes equations. To find the leading eigenvalues and eigenvectors is one of the key issues in the stability analysis. In this study, the Krylov subspace method has been used to develop a general algorithm and then solve the eigenproblem with a large asymmetric matrix in the linearized Navier-Stokes equations. The Krylov subspace method is to construct a set of linearly independent vectors from the linearized equations in which the leading eigenvalues with the largest real parts are included, and orthonormalize the vectors. Then, the Arnoldi's method is used for extracting the desirable eigenvalues with the largest real parts. The advantage of the present method is its ability to calculate a given number of leading eigenvalues including the largest real parts, and corresponding eigenvectors in a large asymmetric matrix. Therefore, this algorithm is general for the h...

The mechanism and the character of singularity caused by eigenvectors are investigated for quasilinear hyperbolic systems.

We study the maximum possible multiplicity of an eigenvalue of a matrix whose graph is a tree, expressing that maximum multiplicity in terms of certain parameters associated with the tree.

In this paper we present results of a comparison of participation factors with a new method that uses the singular value decomposition ( SVD) of complex matrices which are functions of the state matrix A, the input matrix B, the output matrix C and the eigenvalues as measures of modal controllability and observability. The agreement between the two approaches is remarkable. We also give some results about the conditioning of single eigenvalues and eigenvectors and apply them to the same system mentioned above. We find that in some region in the parameter space the eigenvalues corresponding to the area modes approach each other coalescing into a double eigenvalue. At this point and in a region surrounding it in the parameter space the eigenvalues and the eigenvectors become very ill-conditioned rendering the information obtained from the eigenvectors meaningless.

(1998). Computing Eigenvalues and Eigenvectors Without Determinants. Mathematics Magazine: Vol. 71, No. 1, pp. 24-33.

Eigenvalue is an important concept in Linear Algebra. It is well known that the eigenvectors corresponding to different eigenvalues of a square matrix are linear independent. In most of the existing textbooks, this result is proven using mathematical induction. In this note, a new proof using Vandermonde determinant is given. It is shown that this method is much simpler.

Factorization theorems, and properties of sets of eigenvectors, are established for regular selfadjoint quatratic matrix polynomials L(λ) whose leading coefficeint is indefinite or possibly singular, and for which all eigenvalues are real of definite type. The two linear factors obtained have spectra which are just the eigenvalues of L(λ) of positive and negative types, respectively.

Nonlinear matrix equations of the form where Fi, i=0, 1,2 ... ,m are known nxn nonnegative sub-matrices of a state transition matrix arise ubiquitously in the analysis of various stochastic models utilized in queueing, inventory and communication theories. Computation of the minimal nonnegative solution of (1), called the rate matrix R′, is essential for the equilibrium analysis of these models. Previously, this matrix has been computed by iterative techniques. In this paper, an analytical method for the computation of the rate matrix is proposed. Specifically, the eigenvalues of the rate matrix are determined through a Localization Theorem. The eigenvectors and generalized left eigenvectors are then determined to arrive at the Jordan canonical form representation of the rate matrix. Also, the problem of computation of the rate matrix is formulated as a linear programming problem. This method can be adapted in a natural manner for the equilibrium analysis of M/G/l type Markov chains.

This paper deals with closed-loop eigenstructure assignment by output-feedback control in linear multivariable systems for the general case of multiple eigenvalues. The approach is algebraic in nature and operates directly on the closed-loop characteristic equation using fundamental properties of determinants and their derivatives. A compact form for the controller gain matrix is derived and expressed in terms of a set of parameter vectors that specify the non-uniqueness of the solution of the eigenvalue-assignment problem. Some of these parameter vectors assume restricted forms while others are free. A computational scheme is given in which the free parameter vectors are determined in such a way as to assign the corresponding eigenvectors and generalized eigenvectors in a weighted least-square-error sense. An illustrative numerical example is included.

A two-dimensional S /SUB n/ computer code was modified to enable the calculation of a eigenvectors. Entire sets of eigenvectors were determined for sample problems involving extremely coarse mesh structures. A large percentage of the higher harmonics were complex and, for a majority of the cases investigated, the eigenvectors were not complete. For these latter cases the generalized eigenvectors were found, which together with the ordinary eigenvectors do form complete sets. The simple cases analyzed thus far give considerable insight into the eigenvector structure of more complex systems.

In [1], Friedland, Hershkowitz and Schneider introduced a new matrix product called the inflation product and a new class of matrices called inflators. They also proved that the inflation of an arbitrary complex matrix A by an inflator U was rank preserving. In this paper, the definition of inflation is generalized in order to show that a much stronger condition holds: The Jordan canonical form of A××U is obtained from that of A by taking the direct sum of the Jordan canonical form of A and a specified number of 1×1 zero matrices. Further, the eigenvectors and generalized eigenvectors of A××U are completely characterized in terms of a column of U and the generalized eigenvectors for A. Finally, bases for the eigenspaces of inflation-generated matrices are constructed in terms of the corresponding inflation sequence.

(1983). Eigenvalues and eigenvectors of a certain stochastic matrix. Linear and Multilinear Algebra: Vol. 13, No. 4, pp. 345-350.

Formulae for computation of the first- and second-order sensitivity matrices of the eigenvalues and eigenvectors of a matrix with distinct eigenvalues are derived using matrix calculus and the algebra of Kronecker products. The sensitivities of eigenvalues and eigenvectors to all elements of the matrix can thus be expressed by concise matrix equations.

A recursive algorithm for the implicit derivation of the determinant of a symmetric sparse quindiagonal matrix derived from the finite difference discretisation of a self adjoint elliptic partial differential equation in a two-dimensional rectangular domain is developed in terms of its leading principal minors. The algorithm is shown to yield a sequence of polynomials from which the eigenvalues can be obtained by use of the well-known bisection process. Modifications to the inverse iteration method to allow for sparsity of the matrix arrays yields the required eigenvectors.

A method using the simplest matrix generalized inverse, the {1} -inverse, is developed to find a constant state feedback matrix K that yields a set of prescribed eigenvalues for time-invariant linear multivariable systems that contain any finite number of derivatives of the input vector u. The method requires neither controllability assumptions nor the knowledge of eigenvalues and eigenvectors of the plant matrix A. The matrix K exists if a consistency condition for an algebraic linear system of equations is satisfied.

It is a well-known result that if S is a real n X n symmetric matrix then it is diagonalizable, it has n mutually orthogonal eigenvectors, and there exists an orthogonal matrix P such that its columns are eigenvectors of S and such that PTSP = D is the diagonal matrix of corresponding eigenvalues [1, 2, 3, 4, and 5]. In matrix theory courses it is customary to look at a particular real n X n symmetric matrix for some small n > 2 and to try to find its eigenvalues and corresponding eigenvectors. Naturally it is up to an instructor to provide the students with "nice" symmetric matrices, i.e., matrices possessing integer (or rational) eigenvalues and eigenvectors with integer (or rational) components before normalization. One of the goals of this note is to show how to start with an arbitrary orthogonal basis {P 1'P2 * , Pn } of R " and find a real n X n symmetric matrix having the pi for eigenvectors and having either "nice" eigenvalues or having n arbitrarily prescribed real numbers X I, X 2 1 . . . for its eigenvalues. Recall that a real n X n symmetric matrix S is positive (nonnegative) definite if and only if all of its eigenvectors are positive (nonnegative). It is a well-known result that a matrix S is symmetric positive definite if and only if there exists a nonsingular matrix B such that S = BBT [1, 3, and 5]. Another goal here is to sharpen this result and generalize it to characterize real symmetric matrices (Theorems 2 and 3). Our last goal is to provide a simple method for constructing "nice" symmetric matrices. We first present our theorems and then give several examples which illustrate the ideas and methods from the theorems and their proofs. Our first theorem will be useful in two ways; it gives an explicit method for constructing a symmetric matrix with prescribed eigenvectors and eigenvalues and it also is useful in establishing further results. Throughout our discussion, vectors will be considered as column vectors, and we will denote a matrix A with column vectors ai as A = (a1 I a2l ... I a,,).

It is well known that the eigenvectors of a sampled-data system are the same as the eigenvectors of the continuous time system from which the sampled-data system has been derived provided that the eigenvalues of the continuous time system arc distinct. In this paper the eigenvectors of a sampled-data system with confluent eigenvalues arc deduced in terms of the eigenvectors of the continuous time system from which the sampled -data system has boon derived. The theory is illustrated by numerical examples.

Let m be a fixed integer. It is proved that an nxn symmetric random matrix: consisting of mxm blocks has m large in absolute value eigenvalues of order√ n while the remaining eigenvalues are of order fn as n tends to infinity. We assume the associations between observations to be independent random variables i.e. the sample is in some sense infinite dimensional. This random matrix cluster structure-can be obtained by means of the eigenvectors corresponding to the extremal eigenvalues. of the association matrix

The problem of simultaneous assignment of closed-loop eigenvalues and eigenvectors using constant complete state-feedback controller is discussed. It is shown that for a given nth-order multivariable system not necessarily completely controllable, associated with every point. λ in the complex plane, there exists a well-defined subspace (in the n-dimensional vector space) within which the eigenvector v corresponding to the eigenvalue at A must lie in order for a real valued controller. assigning the pair (λ, v), to exist. This sub-space is completely determined by the given system and λ. Based on this fact, a necessary and sufficient condition for k pairs of desired closed-loop eigenvalues-eigenvectors, where 1≤κ≤n, to be simultaneously assignable by state-feedback is proposed and a method for determining a controller to assign them is presented. The method is applicable to the case of linearly independent eigenvectors with distinct or repeated eigenvalues as well as that of multiple eigenvectors

Several theorems are proved showing when a functionally commutative matrix must have a set of constant eigenvectors. Theorems for the converse implication are also given. Counterexamples to both implications in the general case are shown.

This paper looks at an application to the solution of polynomial equations of the power method for finding the eigenvalues of a matrix. This provides useful motivation for the study of eigenvalues and eigenvectors and has the interesting property that it allows regions of the complex plane to be determined in which no undiscovered root can lie.

In this paper we present for the first time an accurate, fast, and easy to implement numerical algorithm to find the eigenvalues and eigenvectors of the equation L(x;λ) = (rx')'+ px + λqx = 0. These ideas follow from a theory of quadratic forms given by the first author and will be applicable in a wide variety of eigenvalue problems. We include test runs to demonstrate that the accuracy of our methods are superior to more conventional projection methods

The infinite zero structure of completely controllable and observable linear multi-variable systems which give rise to square, non-singular, strictly proper transfer function matrices is investigated via the polynomial matrix approach to the solution of the decoupling problem. In the process various connections between infinite zeros, their degrees and geometric and polynomial matrix ideas are demonstrated. The asymptotic behaviour of closed loop eigenvectors under high gain output feedback is also examined.

The problem of pole assignment in a completely controllable linear time-invariant system dx/t = Ax + Bu, y = Cx is considered. A method using matrix generalized inverses is developed for the computation of a matrix K such that the matrix A + BK has prescribed eigenvalues which need satisfy only the condition that a certain number of them are distinct and real; then a feedback law of the form u = r + Kx can be used to achieve the desired pole-placement. The method does not require solution of sets of non-linear equations or manipulation of polynomial matrices, and no knowledge of eigenvalues and/or eigenvectors of A is necessary. If the computed matrix K and the given matrix C satisfy a consistency condition, a matrix Kν such that KνC = K can be directly obtained from K and the desired pole-placement can be realized by an output feedback law u = r + Kνy.

It is known (Porter and Bradshaw 1978 a) that, in the case of self-conjugate distinct eigenvalue spectra, the closed-loop eigenstructure assignable by output feedback is constrained by the requirement that the eigenvectors and reciprocal eigenvectors lie in well-defined subspaces. In this paper a technique is presented which can be used to select the eigenvectors and reciprocal eigenvectors from these subspaces in the case of appropriately augmented (Kimura 1975) controllable and observable continuous-time systems. Thin technique is ideally suited to digital-computer implementation and therefore greatly facilitates the synthesis of both static (Porter and Bradshaw 1978 a) and dynamic (Porter and Bradshaw 1978 b) output-feedback controllers.