On Some “Tame” and “Wild” Aspects of the Problem of Semiscalar Equivalence of Polynomial Matrices

The problem of reducing polynomial matrices to canonical form by using semiscalar equivalent transformations is studied. This problem is wild as a whole. However, it is tame in some special cases. In the paper, classes of polynomial matrices are singled out for which canonical forms with respect to semiscalar equivalence are indicated. We use this tool to construct a canonical form for the families of coefficients corresponding to the polynomial matrices. This form enables one to solve the classification problem for families of numerical matrices up to similarity.     

Similarity transformations of decomposable matrix polynomials and related questions

A relationship is found between the similarity transformations of decomposable matrix polynomials with relatively prime elementary divisors and the equivalence transformations of the corresponding matrices with scalar entries. Matrices with scalar entries are classified with respect to equivalence transformations based on direct sums of lower triangular almost Toeplitz matrices. This solves the similarity problem for a special class of finite matrix sets over the field of complex numbers. Eventually, this problem reduces to the one of special diagonal equivalence between matrices. Invariants of this equivalence are found.     

Solvability of matrix equations in rings of quasi-diagonal matrices and similarity of matrix polynomials

A certain standard form is found for a complex matrix with respect to equivalent transformations by quasi-diagonal matrices. The solvability of certain matrix equations in the rings of quasi-diagonal matrices is examined using this standard form.     

Generating equations approach for quadratic matrix equations

We show how Van Loan's method for annulling the (2,1) block of skew‐Hamiltonian matrices by symplectic‐orthogonal similarity transformation generalizes to general matrices and provides a numerical algorithm for solving the general quadratic matrix equation: For skew‐Hamiltonian matrices we find their canonical form under a similarity transformation and find the class of all symplectic‐orthogonal similarity transformations for annulling the (2,1) block and simultaneously bringing the (1,1) block to Hessenberg form. We present a structure‐preserving algorithm for the solution of continuous‐time algebraic Riccati equation. Unlike other methods in the literature, the final transformed Hamiltonian matrix is not in Hamiltonian–Schur form. Three applications are presented: (a) for a special system of partial differential equations of second order for a single unknown function, we obtain the matrix of partial derivatives of second order of the unknown function by only algebraic operations and differentiation of functions; (b) for a similar transformation of a complex matrix into a symmetric (and three‐diagonal) one by applying only finite algebraic transformations; and (c) for finite‐step reduction of the eigenvalues–eigenvectors problem of a Hermitian matrix to the eigenvalues– eigenvectors problem of a real symmetric matrix of the same dimension. Copyright © 1999 John Wiley & Sons, Ltd.     

A compound matrix algorithm for the computation of the Smith form of a polynomial matrix

In the present paper is presented a numerical method for the exact reduction of a singlevariable polynomial matrix to its Smith form without finding roots and without applying unimodular transformations. Using the notion of compound matrices, the Smith canonical form of a polynomial matrixM(s)^nxn[s] is calculated directly from its definition, requiring only the construction of all thep-compound matricesC _p (M(s)) ofM(s), 1<pn. This technique produces a stable and accurate numerical algorithm working satisfactorily for any polynomial matrix of any degree.     

Reachability of condensed forms via unitary congruence transformations

There are several well-known facts about unitary similarity transformations of complex n-by-n matrices: every matrix of order n = 3 can be brought to tridiagonal form by a unitary similarity transformation; if n ≥ 5, then there exist matrices that cannot be brought to tridiagonal form by a unitary similarity transformation; for any fixed set of positions (pattern) S whose cardinality exceeds n(n ? 1)/2, there exists an n-by-n matrix A such that none of the matrices that are unitarily similar to A can have zeros in all of the positions in S. It is shown that analogous facts are valid if unitary similarity transformations are replaced by unitary congruence ones.     

Semiscalar equivalence and the factorization of polynomial matrices

One considers the problem of the factorization of polynomial matrices over an arbitrary field in connection with their reducibility by semiscalar equivalent transformations to triangular form with the invariant factors along the principal diagonal. In particular, one establishes a criterion for the representability of a polynomial matrix in the form of a product of factors (the first of which is unital), the product of the canonical diagonal forms of which is equal to the canonical diagonal form of the given matrix. There is given also a method for the construction of such factorizations.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 5, pp. 644–649, May, 1990.     

A Schur decomposition for Hamiltonian matrices

A Schur-type decomposition for Hamiltonian matrices is given that relies on unitary symplectic similarity transformations. These transformations preserve the Hamiltonian structure and are numerically stable, making them ideal for analysis and computation. Using this decomposition and a special singular-value decomposition for unitary symplectic matrices, a canonical reduction of the algebraic Riccati equation is obtained which sheds light on the sensitivity of the nonnegative definite solution. After presenting some real decompositions for real Hamiltonian matrices, we look into the possibility of an orthogonal symplectic version of the QR algorithm suitable for Hamiltonian matrices. A finite-step initial reduction to a Hessenberg-type canonical form is presented. However, no extension of the Francis implicit-shift technique was found, and reasons for the difficulty are given.     

Fast Givens rotations for orthogonal similarity transformations

Summary Fast Givens rotations with half as many multiplications are proposed for orthogonal similarity transformations and a matrix notation is introduced to describe them more easily. Applications are proposed and numerical results are examined for the Jacobi method, the reduction to Hessenberg form and the QR-algorithm for Hessenberg matrices. It can be seen that in general fast Givens rotations are competitive with Householder reflexions and offer distinct advantages for sparse matrices.     

Monotone additive matrix transformations

We investigate additive transformations on the space of real or complex matrices that are monotone with respect to any admissible partial order relation. A complete characterization of these transformations is obtained. In the real case, we show that such transformations are linear and that all nonzero monotone transformations are bijective. As a corollary, we characterize all additive transformations that are monotone with respect to certain classical matrix order relations, in particular, with respect to the Drazin order, left and right *-orders, and the diamond order.     

On the reduction of a set of numerical matrices to quasi-diagonal form with one similarity transformation

We consider finite sets of numerical matrices and the polynomial matrices corresponding to them that have the Smith form diag (1, (x), ..., (x)). We solve the problem of reducing such sets to canonical form with one similarity transformation assuming that all the roots of the invariant polynomial (x) are simple.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 4, 1997, pp. 101–105.     

A matrix pencil approach computing the elementary divisors of a matrix: Numerical aspects and applications

In the present paper is presented a new matrix pencil-based numerical approach achieving the computation of the elementary divisors of a given matrixA ∈ C ^{n × n}. This computation is attained without performing similarity transformations and the whole procedure is based on the construction of the Piecewise Arithmetic Progression Sequence (PAPS) of the associated pencil λI _n - A of matrix A, for all the appropriate values of λ belonging to the set of eigenvalues of A. This technique produces a stable and accurate numerical algorithm working satisfactorily for matrices with a well defined eigenstructure. The whole technique can be applied for the computation of the first, second and Jordan canonical form of a given matrixA ∈ C ^{n × n}. The results are accurate for matrices possessing a well defined canonical form. In case of defective matrices, indications of the most appropriately computed canonical form are given.     

Classification of multivariate skew polynomial rings over finite fields via affine transformations of variables

In this work, free multivariate skew polynomial rings are considered, together with their quotients over ideals of skew polynomials that vanish at every point (which includes minimal multivariate skew polynomial rings). We provide a full classification of such multivariate skew polynomial rings (free or not) over finite fields. To that end, we first show that all ring morphisms from the field to the ring of square matrices are diagonalizable, and that the corresponding derivations are all inner derivations. Secondly, we show that all such multivariate skew polynomial rings over finite fields are isomorphic as algebras to a multivariate skew polynomial ring whose ring morphism from the field to the ring of square matrices is diagonal, and whose derivation is the zero derivation. Furthermore, we prove that two such representations only differ in a permutation on the field automorphisms appearing in the corresponding diagonal. The algebra isomorphisms are given by affine transformations of variables and preserve evaluations and degrees. In addition, ours proofs show that the simplified form of multivariate skew polynomial rings can be found computationally and explicitly.     

Determining conformal transformations in from minimal correspondence data

In this paper, we derive a method to determine a conformal transformation in n‐dimensional Euclidean space in closed form given exact correspondences between data. We show that a minimal data set needed for correspondence is a localized vector frame and an additional point. In order to determine the conformal transformation, we use the representation of the conformal model of geometric algebra by extended Vahlen matrices— 2 ×2 matrices with entries from Euclidean geometric algebra (the Clifford algebra of ). This reduces the problem on the determination of a Euclidean orthogonal transformation from given vector correspondences, for which solutions are known. We give a closed form solution for the general case of conformal (in contrast, anti‐conformal) transformations, which preserve (in contrast, reverse) angles locally, as well as for the important special case when it is known that the conformal transformation is a rigid body motion—also known as a Euclidean transformation—which additionally preserves Euclidean distances. Copyright © 2011 John Wiley & Sons, Ltd.     

Classification of squared normal operators on unitary and Euclidean spaces

We give a canonical form for a complex matrix whose square is normal under transformations of unitary similarity as well as a canonical form for a real matrix whose square is normal under transformations of orthogonal similarity. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 4, pp. 225–232, 2007.     

論矩陣的一種變換

<正> 在本文中,數域限定為複數域.我們要來研究如下的變換:(1)(它將方陣A變成方陣B),式中P表示任意正則陣,P表示P的元素的共軛救構成的陣.所有的變换(1)顯然成羣.這種變換現在姑稱之為種變換.如果二方陣A與B可由一個種變換變此成彼,我們就說,A與B是對相似的.     

An implicit QR algorithm for symmetric semiseparable matrices

The QR algorithm is one of the classical methods to compute the eigendecomposition of a matrix. If it is applied on a dense n × n matrix, this algorithm requires O(n³) operations per iteration step. To reduce this complexity for a symmetric matrix to O(n), the original matrix is first reduced to tridiagonal form using orthogonal similarity transformations. In the report (Report TW360, May 2003) a reduction from a symmetric matrix into a similar semiseparable one is described. In this paper a QR algorithm to compute the eigenvalues of semiseparable matrices is designed where each iteration step requires O(n) operations. Hence, combined with the reduction to semiseparable form, the eigenvalues of symmetric matrices can be computed via intermediate semiseparable matrices, instead of tridiagonal ones. The eigenvectors of the intermediate semiseparable matrix will be computed by applying inverse iteration to this matrix. This will be achieved by using an O(n) system solver, for semiseparable matrices. A combination of the previous steps leads to an algorithm for computing the eigenvalue decompositions of semiseparable matrices. Combined with the reduction of a symmetric matrix towards semiseparable form, this algorithm can also be used to calculate the eigenvalue decomposition of symmetric matrices. The presented algorithm has the same order of complexity as the tridiagonal approach, but has larger lower order terms. Numerical experiments illustrate the complexity and the numerical accuracy of the proposed method. Copyright © 2005 John Wiley & Sons, Ltd.     

Normalizing and variance stabilizing transformations of multivariate statistics under an elliptical population

Summary Normalizing and variance stabilizing transformations of a sample correlation, multiple correlation and canonical correlation coefficients are obtained under an elliptical population. It is shown that the Fisher'sz-transformation is efficient for these statistics. A normalizing transformation is also studied for a latent root of a sample covariance matrix in an elliptical sample.     

Polynomial numerical hulls of matrix polynomials,II

In this article, we study some algebraic and geometrical properties of polynomial numerical hulls of matrix polynomials and joint polynomial numerical hulls of a finite family of matrices (possibly the coefficients of a matrix polynomial). Also, we study polynomial numerical hulls of basic A-factor block circulant matrices. These are block companion matrices of particular simple monic matrix polynomials. By studying the polynomial numerical hulls of the Kronecker product of two matrices, we characterize the polynomial numerical hulls of unitary basic A-factor block circulant matrices.     

Semiscalar equivalence and the smith normal form of polynomial matrices

We give conditions under which a set of polynomial matrices over a finite field can be simultaneously reduced by means of semiscalar equivalent transformations to a special triangular form with invariant factors on the principal diagonals. We investigate multiplicative properties of the Smith normal form of polynomial matrices and in particular we identify a class of polynomial matrices for which the Smith normal form of the product matrix is equal to the product of the Smith normal forms of the factor matrices.Translated from Matematicheskie Metody i Fiziko-mekhanicheskie Polya, No. 26, pp. 13–16, 1987.