Jordan structures of alternating matrix polynomials

Alternating matrix polynomials, that is, polynomials whose coefficients alternate between symmetric and skew-symmetric matrices, generalize the notions of even and odd scalar polynomials. We investigate the Smith forms of alternating matrix polynomials, showing that each invariant factor is an even or odd scalar polynomial. Necessary and sufficient conditions are derived for a given Smith form to be that of an alternating matrix polynomial. These conditions allow a characterization of the possible Jordan structures of alternating matrix polynomials, and also lead to necessary and sufficient conditions for the existence of structure-preserving strong linearizations. Most of the results are applicable to singular as well as regular matrix polynomials.     

Normal forms in matrix spaces

For the groupGL(m, C)xGL(n, C) acting on the space ofmxn matrices over C, we introduce a class of subgroups which we call admissible. We suggest an algorithm to reduce an arbitrary matrix to a normal form with respect to an action of any admissible group. This algorithm covers various classification problems, including the wild problem of bringing a pair of matrices to normal form by simultaneous similarity. The classical left, right, two-sided and similarity transformations turns out to be admissible. However, the stabilizers of known normal forms (Smith's, Jordan's), generally speaking, are not admissible, and this obstructs inductive steps of our algorithm. This is the reason that we introduce modified normal forms for classical actions.Partially supported by Israel Science Foundation     

An extremal sparsity property of the Jordan canonical form

We prove that among all the matrices that are similar to a given square complex matrix, the Jordan canonical form has the largest number of off-diagonal zero entries. We also characterize those matrices that attain this largest number.     

The {\cal {D Q R}} algorithm, basic theory, convergence, and conditional stability

Summary. We describe a fast matrix eigenvalue algorithm that uses a matrix factorization and reverse order multiply technique involving three factors and that is based on the symmetric matrix factorization as well as on –orthogonal reduction techniques where is computed from the given matrix . It operates on a similarity reduction of a real matrix to general tridiagonal form and computes all of 's eigenvalues in operations, where the part of the operations is possibly performed over , instead of the 7–8 real flops required by the eigenvalue algorithm. Potential breakdo wn of the algorithm can occur in the reduction to tridiagonal form and in the –orthogonal reductions. Both, however, can be monitored during the computations. The former occurs rather rarely for dimensions and can essentially be bypassed, while the latter is extremely rare and can be bypassed as well in our conditionally stable implementation of the steps. We prove an implicit theorem which allows implicit shifts, give a convergence proof for the algorithm and show that is conditionally stable for general balanced tridiagonal matrices . Received April 25, 1995 / Revised version received February 9, 1996     

An algorithm for determining if the inverse of a strictly diagonally dominant Stieltjes matrix is strictly ultrametric

Summary It was recently shown that the inverse of a strictly ultrametric matrix is a strictly diagonally dominant Stieltjes matrix. On the other hand, as it is well-known that the inverse of a strictly diagonally dominant Stieltjes matrix is a real symmetric matrix with nonnegative entries, it is natural to ask, conversely, if every strictly diagonally dominant Stieltjes matrix has a strictly ultrametric inverse. Examples show, however, that the converse is not true in general, i.e., there are strictly diagonally dominant Stieltjes matrices in ^n×n (for everyn3) whose inverses are not strictly ultrametric matrices. Then, the question naturally arises if one can determine which strictly diagonally dominant Stieltjes matrices, in ^n×n (n3), have inverses which are strictly ultrametric. Here, we develop an algorithm, based on graph theory, which determines if a given strictly diagonally dominant Stieltjes matrixA has a strictly ultrametric inverse, where the algorithm is applied toA and requires no computation of inverse. Moreover, if this given strictly diagonally dominant Stieltjes matrix has a strictly ultrametric inverse, our algorithm uniquely determines this inverse as a special sum of rank-one matrices.Research supported by the National Science FoundationResearch supported by the Deutsche Forschungsgemeinschaft     

Singular, nonsingular, and bounded rank completions of ACI-matrices

An affine column independent matrix is a matrix whose entries are polynomials of degree at most 1 in a number of indeterminates where no indeterminate appears with a nonzero coefficient in two different columns. A completion is a matrix obtained by giving values to each of the indeterminates. Affine column independent matrices are more general than partial matrices where each entry is either a constant or a distinct indeterminate. We determine when the rank of all completions of an affine column independent matrix is bounded by a given number, generalizing known results for partial matrices. We also characterize the square partial matrices over a field all of whose completions are nonsingular. The maximum number of free entries in such matrices of a given order is determined as well as the partial matrices with this maximum number of free entries.     

Linear complexity algorithm for semiseparable matrices

A new linear complexity algorithm for general nonsingular semiseparable matrices is presented. For symmetric matrices whose semiseparability rank equals to 1 this algorithm leads to an explicit formula for the inverse matrix.Supported in part by the NSF Grant DMS 9306357     

On the closed representation for the inverses of Hessenberg matrices

The general representation for the elements of the inverse of any Hessenberg matrix of finite order is here extended to the reduced case with a new proof. Those entries are given with proper Hessenbergians from the original matrix. It justifies both the use of linear recurrences of unbounded order for such computations on matrices of intermediate order, and some elementary properties of the inverse. These results are applied on the resolvent matrix associated to a finite Hessenberg matrix in standard form. Two examples on the unit disk are given.     

Permanents of doubly stochastic matrices with fixed zero pattern

The doubly stochastic matrices with a given zero pattern which are closest in Euclidean norm to J_nn, the matrix with each entry equal to 1/n, are identified. If the permanent is restricted to matrices having a given zero pattern confined to one row or to one column, the permanent achieves a local minimum at those matrices with that zero pattern which are closest to J_nn. This need no longer be true if the zeros lie in more than one row or column.     

Generalized even and odd totally positive matrices

A generalization of the definition of an oscillatory matrix based on the theory of cones is given in this paper. The positivity and simplicity of all the eigenvalues of a generalized oscillatory matrix are proved. Classes of generalized even and odd oscillatory matrices are introduced. Spectral properties of the obtained matrices are studied. Criteria of generalized even and odd oscillation are given. Examples of generalized even and odd oscillatory matrices are presented.     

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.     

Detecting hyperbolic and definite matrix polynomials

Hyperbolic or more generally definite matrix polynomials are important classes of Hermitian matrix polynomials. They allow for a definite linearization and can therefore be solved by a standard algorithm for Hermitian matrices. They have only real eigenvalues which can be characterized as minmax and maxmin values of Rayleigh functionals, but there is no easy way to test if a given polynomial is hyperbolic or definite or not. Taking advantage of the safeguarded iteration which converges globally and monotonically to extreme eigenvalues we obtain an efficient algorithm that identifies hyperbolic or definite polynomials and enables the transformation to an equivalent definite linear pencil. Numerical examples demonstrate the efficiency of the approach.     

Unitary matrix digraphs and minimum semidefinite rank

For an undirected simple graph G, the minimum rank among all positive semidefinite matrices with graph G is called the minimum semidefinite rank (msr) of G. In this paper, we show that the msr of a given graph may be determined from the msr of a related bipartite graph. Finding the msr of a given bipartite graph is then shown to be equivalent to determining which digraphs encode the zero/nonzero pattern of a unitary matrix. We provide an algorithm to construct unitary matrices with a certain pattern, and use previous results to give a lower bound for the msr of certain bipartite graphs.     

Multiple LU factorizations of a singular matrix

A singular matrix A may have more than one LU factorizations. In this work the set of all LU factorizations of A is explicitly described when the lower triangular matrix L is nonsingular. To this purpose, a canonical form of A under left multiplication by unit lower triangular matrices is introduced. This canonical form allows us to characterize the matrices that have an LU factorization and to parametrize all possible LU factorizations. Formulae in terms of quotient of minors of A are presented for the entries of this canonical form.     

On schur complements in an ep matrix

It is established that under certain conditions a Schur complement in an EP matrix is as well an EP matrix. As an application a decomposition of a partitioned matrix into a sum of EP matrices is given.     

Fischer determinantal inequalities and Highamʼs Conjecture

We obtain a Fischer type determinantal inequality for matrices with given angular numerical range. We discuss the growth factor for Gaussian elimination for linear systems in which the coefficient matrix has this form and give a proof of Higham?s Conjecture.     

由Smith标准形求解矩阵方程∑AiXBi=C

By using the Smith normal form of polynomial matrix and algebraic methods, this paper discusses the solvability for the linear matrix equation ∑AiXBi = C over a field, and obtains the explicit formulas of general solution or unique solution.     

Canonical forms for normal matrices that commute with their complex conjugate

We consider the class of normal complex matrices that commute with their complex conjugate. We show that such matrices are real orthogonally similar to a canonical direct sum of 1-by-1 and certain 2-by-2 matrices. A canonical form for quasi-real normal matrices is obtained as a special case. We also exhibit a special form of the spectral theorem for normal matrices that commute with their conjugate.     

Inversion of regular analytic matrix functions: Local Smith form and subspace duality

This paper analyzes the relation between the local rank-structure of a regular analytic matrix function and the one of its inverse function. The local rank factorization (lrf) of a matrix function is introduced, which characterizes extended canonical systems of root functions and the local Smith form. An interpretation of the local rank factorization in terms of Jordan chains and Jordan pairs is provided. Duality results are shown to hold between the subspaces associated with the lrf of the matrix function and the one of its reduced adjoint.