Residues of codimension one singular holomorphic distributions

For a codimension one locally-free singular holomorphic distribution, we give a residue formula in terms of the conormal sheaf given by Pfaffian equations. We also prove a Baum-Bott type residue formula for singular distributions.      

Computing the generalized singular values/vectors of large sparse or structured matrix pairs

Summary. We present a numerical algorithm for computing a few extreme generalized singular values and corresponding vectors of a sparse or structured matrix pair . The algorithm is based on the CS decomposition and the Lanczos bidiagonalization process. At each iteration step of the Lanczos process, the solution to a linear least squares problem with as the coefficient matrix is approximately computed, and this consists the only interface of the algorithm with the matrix pair . Numerical results are also given to demonstrate the feasibility and efficiency of the algorithm. Received April 1, 1994 / Revised version received December 15, 1994     

F-singularities of pairs and Inversion of Adjunction of arbitrary codimension

We generalize the notions of F-regular and F-pure rings to pairs of rings R and ideals with real exponent t>0, and investigate these properties. These F-singularities of pairs correspond to singularities of pairs of arbitrary codimension in birational geometry. Via this correspondence, we prove a sort of Inversion of Adjunction of arbitrary codimension, which states that for a pair (X,Y) of a smooth variety X and a closed subscheme , if the restriction (Z,Y|_Z) to a normal -Gorenstein closed subvariety is klt (resp. lc), then the pair (X,Y+Z) is plt (resp. lc) near Z.     

The product-product singular value decomposition of matrix triplets

A new decomposition of a matrix triplet (A, B, C) corresponding to the singular value decomposition of the matrix productABC is developed in this paper, which will be termed theProduct-Product Singular Value Decomposition (PPSVD). An orthogonal variant of the decomposition which is more suitable for the purpose of numerical computation is also proposed. Some geometric and algebraic issues of the PPSVD, such as the variational and geometric interpretations, and uniqueness properties are discussed. A numerical algorithm for stably computing the PPSVD is given based on the implicit Kogbetliantz technique. A numerical example is outlined to demonstrate the accuracy of the proposed algorithm.The work was partially supported by NSF grant DCR-8412314.     

The generalized inverse of a perturbed singular matrix

It is shown that, whenA is singular, (A+A ₁)^–1 can be expanded into a Laurent's series in. The coefficients of the expansion are given in an explicit form. The case where A+A ₁ vanishes identically in is also studied and a generalized inverse ofA+A ₁ is given.

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Zusammenfassung Es wird gezeigt, daß wennA singulär ist (A+A ₁)^–1 auf eine Laurent-Serie in erweitert werden kann. Die Koeffizienten der Erweiterung werden in expliziter Form gegeben. Der Fall wo A+A ₁ sich identisch in auflöst wird ebenfalls untersucht und eine verallgemeinerte Umkehrung vonA+A ₁ wird angegeben.

     

Analysis of singular matrix pencils

One considers the spectral problem for a singular pencil D(λ)=A + λB of matrices A and B (A and B are rectangular matrices or det D(λ)≡0). One represents an algorithm which allows us to find the reducing subspaces for D(λ) and with their aid to reduce the dimension of the initial pencil, by isolating from it the zero block, the blocks corresponding to the right and left polynomial solutions of the equations (A+λB)x(λ)=0 and y(λ)(A+λB)=0, respectively, as well as the block corresponding to the regular kernel of the pencil D(λ). The algorithm is based on the application of the normalized process which uses the numerically stable elementary orthogonal transformations (the matrices of plane rotations or reflections).     

Factorization of a rational matrix: The singular case

We study the problem of minimal factorization of an arbitrary rational matrix R(), i. e. where R() is not necessarily square or invertible. Following the definition of minimality used here, we show that the problem can be solved via a generalized eigenvalue problem which will be singular when R() is singular. The concept of invariant subspace, which has been used in the solution of the minimal factorization problem for regular matrices, is now replaced by a reducing subspace, a recently introduced concept which is a logical extension of invariant and deflating subspaces to the singular pencil case.     

The Drazin inverse of a singular,unreduced tridiagonal matrix

For a tridiagonal, singular matrix A   we present a method for the computation of the polynomial p(λ)$p(\lambda )$ such that A^D=p(A)$A^D=p(A)$ holds, where A^D$A^D$ is the Drazin inverse of A. The approach is based on the recursion of characteristic polynomials of leading principal submatrices of A.     

Computing the nearest stable matrix pairs

In this paper, we study the nearest stable matrix pair problem: given a square matrix pair (E,A), minimize the Frobenius norm of (Δ_E,Δ_A) such that (E+Δ_E,A+Δ_A) is a stable matrix pair. We propose a reformulation of the problem with a simpler feasible set by introducing dissipative Hamiltonian matrix pairs: A matrix pair (E,A) is dissipative Hamiltonian if A=(J−R)Q with skew‐symmetric J, positive semidefinite R, and an invertible Q such that Q^TE is positive semidefinite. This reformulation has a convex feasible domain onto which it is easy to project. This allows us to employ a fast gradient method to obtain a nearby stable approximation of a given matrix pair.     

The codimension of degenerate pencils

Let d_n[d_n(r)] denote the codimension of the set of pairs of n×n Hermitian [really symmetric] matrices (A, B) for which det(λI?A?xB)=p(λ,x) is a reducible polynomial. We prove that d_n(r)?n?1, d_n?_n?1 (n odd), d_n?n (n even). We conjecture that the equality holds in all three inequalities. We prove this conjecture for n=2,3.     

Computing the optimal commuting matrix pairs

A matrix can be modified by an additive perturbation so that it commutes with any given matrix. In this paper, we discuss several algorithms for computing the smallest perturbation in the Frobenius norm for a given matrix pair. The algorithms have applications in 2-D direction-of-arrival finding in array signal processing. The work of first author was supported in part by NSF grant CCR-9308399. The work of the second author was supported in part by China State Major Key Project for Basic Researches.     

On singular matrix variate beta distribution

1.IntrodnctionThispaperextendsthestudyofthesingularmatrixvariatebetadistributionofrank1[1]tothecaseofageneralrank.Astherelateddistributiontonormalsampling,thematrixvariatebetadistribution(alsocalledthemultivariatebetadistribution)hasbeenstudiedextens...     

The semigroup generated by a unitary orbit of a singular matrix

We investigate the structure of the multiplicative semigroup generated by the set of matrices that are unitarily equivalent to a given singular matrix A. In particular, we give necessary and sufficient conditions, in terms of the singular values of A, for such a semigroup to consist of all matrices of rank not exceeding the rank of A.     

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.     

The semigroup generated by a unitary orbit of a singular matrix

We investigate the structure of the multiplicative semigroup generated by the set of matrices that are unitarily equivalent to a given singular matrix A. In particular, we give necessary and sufficient conditions, in terms of the singular values of A, for such a semigroup to consist of all matrices of rank not exceeding the rank of A.     

Smallest singular value of a random rectangular matrix

We prove an optimal estimate of the smallest singular value of a random sub‐Gaussian matrix, valid for all dimensions. For an N × n matrix A with independent and identically distributed sub‐Gaussian entries, the smallest singular value of A is at least of the order √N ? √n ? 1 with high probability. A sharp estimate on the probability is also obtained. © 2009 Wiley Periodicals, Inc.