Generalized canonical factorization of matrix and operator functions with definite hermitian part

It is proved that rational matrix functions with definite hermitian part on the real line admit a generalized canonical factorization. The functions are allowed to have poles on the real line. A generalization of this result to a class of operator functions is obtained as well.Partially supported by an NSF grant     

The Albanese map of a 3-fold of general type whose canonical map is composed with a pencil

Received: 15 October 1999; in final form: 13 June 2000 / Published online: 29 April 2002     

A matrix pencil approach to the existence of compactly supported reconstruction functions in average sampling

The aim of this work is to solve a question raised for average sampling in shift-invariant spaces by using the well-known matrix pencil theory. In many common situations in sampling theory, the available data are samples of some convolution operator acting on the function itself: this leads to the problem of average sampling, also known as generalized sampling. In this paper we deal with the existence of a sampling formula involving these samples and having reconstruction functions with compact support. Thus, low computational complexity is involved and truncation errors are avoided. In practice, it is accomplished by means of a FIR filter bank. An answer is given in the light of the generalized sampling theory by using the oversampling technique: more samples than strictly necessary are used. The original problem reduces to finding a polynomial left inverse of a polynomial matrix intimately related to the sampling problem which, for a suitable choice of the sampling period, becomes a matrix pencil. This matrix pencil approach allows us to obtain a practical method for computing the compactly supported reconstruction functions for the important case where the oversampling rate is minimum. Moreover, the optimality of the obtained solution is established.     

Coding with canonical functions

A function f from ω₁ to the ordinals is called a canonical function for an ordinal α if f represents α in any generic ultrapower induced by forcing with $\wp ({\omega }_1)/{\mathrm{NS}}_{{\omega }_1}$. We introduce here a method for coding sets of ordinals using canonical functions from ω₁ to ω₁. Combining this approach with arguments from 3 , we show, assuming the Continuum Hypothesis, that for each cardinal κ there is a forcing construction preserving cardinalities and cofinalities forcing that every subset of κ is an element of the inner model $\mathbf{L}(\wp ({\omega }_1))$.     

Asymptotic distributions of functions of the eigenvalues of sample covariance matrix and canonical correlation matrix in multivariate time series

Let S = (1/n) Σ_t=1ⁿ X(t) X(t)′, where X(1), …, X(n) are p × 1 random vectors with mean zero. When X(t) (t = 1, …, n) are independently and identically distributed (i.i.d.) as multivariate normal with mean vector 0 and covariance matrix Σ, many authors have investigated the asymptotic expansions for the distributions of various functions of the eigenvalues of S. In this paper, we will extend the above results to the case when {X(t)} is a Gaussian stationary process. Also we shall derive the asymptotic expansions for certain functions of the sample canonical correlations in multivariate time series. Applications of some of the results in signal processing are also discussed.     

Solution of the partial eigenvalue problem for a regular matrix pencil

Algorithms are proposed for the solution of the partial eigenvalue problem for regular polynomial matrix pencils. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 159, pp. 33–39, 1987.     

On surfaces with a canonical pencil

We classify the minimal surfaces of general type with K ² ≤ 4χ ? 8 whose canonical map is composed with a pencil, up to a finite number of families. More precisely we prove that there is exactly one irreducible family for each value of ${\chi \gg 0,\,4\chi-10 \leq K^2 \leq 4\chi-8}$ . All these surfaces are complete intersections in a toric 4-fold and bidouble covers of Hirzebruch surfaces. The surfaces with K ² = 4χ ? 8 were previously constructed by Catanese as bidouble covers of ${\mathbb{P}^1 \times \mathbb{P}^1}$ .     

On canonical factorization of rational matrix functions

This paper concerns the problem of canonical factorization of a rational matrix functionW() which is analytic but may benot invertible at infinity. The factors are obtained explicitly in terms of the realization of the original matrix function. The cases of symmetric factorization for selfadjoint and positive rational matrix functions are considered separately.     

The computation of Kronecker's canonical form of a singular pencil

We develop stable algorithms for the computation of the Kronecker structure of an arbitrary pencil. This problem can be viewed as a generalization of the well-known eigenvalue problem of pencils of the type λI?A. We first show that the elementary divisors (λ ? α)ⁱ of a regular pencil λB?A can be retrieved with a deflation algorithm acting on the expansion (λ ? α)B ? (A ? αB). This method is a straightforward generalization of Kublanovskaya's algorithm for the determination of the Jordan structure of a constant matrix. We also show how to use this method to determine the structure of the infinite elementary divisors of λB?A. In the case of singular pencils, the occurrence of Kronecker indices—containing the singularity of the pencil—somewhat complicates the problem. Yet our algorithm retrieves these indices with no additional effort, when determining the elementary divisors of the pencil. The present ideas can also be used to separate from an arbitrary pencil a smaller regular pencil containing only the finite elementary divisors of the original one. This is shown to be an effective tool when used together with the QZ algorithm.     

An algorithm for computing the spectral structure of a singular linear matrix pencil

An algorithm is proposed for computing the structure of the Kronecker canonical form for a singular linear matrix pencil. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Institute im. V. A. Steklova AN SSSR, Vol. 159, pp. 23–32, 1987.     

The spectral property of dual matrix strings as a consequence of the existence of spectral matrix functions for canonical systems

The term dual string for scalar strings was introduced in [KK1], where some connections between the spectra of a string and its dual were studied. In [KK2] it was shown that if () is a spectral function of a scalar stringS ₁ with nonnegative spectrum (in the sense of [KK2]), then the function

The characteristic polynomial of a principal subpencil of a Hermitian matrix pencil

Given n-square Hermitian matrices A,B, let A_i,B_i denote the principal (n?1)- square submatrices of A,B, respectively, obtained by deleting row i and column i. Let μ, λ be independent indeterminates. The first main result of this paper is the characterization (for fixed i) of the polynomials representable as det(μA_i?λB_i) in terms of the polynomial det(μA?λB) and the elementary divisors, minimal indices, and inertial signatures of the pencil μA?λB. This result contains, as a special case, the classical interlacing relationship governing the eigenvalues of a principal sub- matrix of a Hermitian matrix. The second main result is the determination of the number of different values of i to which the characterization just described can be simultaneously applied.     

Double piling structure of matrix monotone functions and of matrix convex functions

There are basic equivalent assertions known for operator monotone functions and operator convex functions in two papers by Hansen and Pedersen. In this note we consider their results as correlation problem between two sequences of matrix n-monotone functions and matrix n-convex functions, and we focus the following three assertions at each label n among them:

(i) f(0)0 and f is n-convex in [0,α),

(ii) For each matrix a with its spectrum in [0,α) and a contraction c in the matrix algebra M_n,

f(cac)cf(a)c,