Pietsch's factorization theorem for dominated polynomials

We prove that, like in the linear case, there is a canonical prototype of a p-dominated homogeneous polynomial through which every p-dominated polynomial between Banach spaces factors.     

On the factorization of matrix polynomials with commuting coefficients

We study the decomposability of a regular matrix polynomial A(x)=A₀x³+A₁x^s–7+.+A_s with commuting coefficients into factors under the assumption that some coefficient A_t has one of the following properties: A_t has only one elementary divisor; all the characteristic roots of the matrix A_t have multiplicity at most two.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 32, 1990, pp. 20–23.     

Integrable Lagrangian correspondences and the factorization of matrix polynomials

M. V. Lomonosov Moscow State University. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 25, No. 2, pp. 38–49, April–June, 1991.     

A generalization of the inertia theorem for quadratic matrix polynomials

We show that the inertia of a quadratic matrix polynomial is determined in terms of the inertia of its coefficient matrices if the leading coefficient is Hermitian and nonsingular, the constant term is Hermitian, and the real part of the coefficient matrix of the first degree term is definite. In particular, we prove that the number of zero eigenvalues of such a matrix polynomial is the same as the number of zero eigenvalues of its constant term. We also give some new results for the case where the real part of the coefficient matrix of the first degree term is semidefinite.     

A generalized Cayley-Hamilton theorem for polynomials with matrix coefficients

A family F of square matrices of the same order is called a quasi-commuting family if (AB-BA)C=C(AB-BA) for all A,B,C∈F where A,B,C need not be distinct. Let f_k(x₁,x₂,…,x_p),(k=1,2,…,r), be polynomials in the indeterminates x₁,x₂,…,x_p with coefficients in the complex field C, and let M₁,M₂,…,M_r be n×n matrices over C which are not necessarily distinct. Let and let δ_F(x₁,x₂,…,x_p)=detF(x₁,x₂,…,x_p). In this paper, we prove that, for n×n matrices A₁,A₂,…,A_p over C, if {A₁,A₂,…,A_p,M₁,M₂,…,M_r} is a quasi-commuting family, then F(A₁,A₂,…,A_p)=O implies that δ_F(A₁,A₂,…,A_p)=O.     

Rakhmanov's theorem for orthogonal matrix polynomials on the unit circle

Rakhmanov's theorem for orthogonal polynomials on the unit circle gives a sufficient condition on the orthogonality measure for orthogonal polynomials on the unit circle, in order that the reflection coefficients (the recurrence coefficients in the Szegő recurrence relation) converge to zero. In this paper we give the analog for orthogonal matrix polynomials on the unit circle.     

A factorization theorem for matrices

It is shown that a nonscalar invertible square matrix can be written as a product of two square matrices with prescribed eigenvalues subject only to the obvious determinant condition. As corollaries, we give short proofs of some known results such as Ballantine's characterization of products of four or five positive definite matrices, the commutator theorem of Shoda-Thompson for fields with sufficiently many elements and other results.     

Holomorphic factorization of polynomials

We give necessary and sufficient conditions for a holomorphic factorization of an irreducible polynomial P(s, λ), s ∈ Cⁿ, λ ∈ C, in a domain Ω ? Cⁿ which is connected with the ordering of the real part of the roots of the equation P(s, λ) = 0, s ∈ Ω.     

A factorization theorem for banded matrices

In this paper we prove a factorization theorem for strictly m-banded totally positive matrices. We show that such a matrix is a product of m one-banded matrices with positive entries.     

A unique factorization theorem for matroids

We study the combinatorial, algebraic and geometric properties of the free product operation on matroids. After giving cryptomorphic definitions of free product in terms of independent sets, bases, circuits, closure, flats and rank function, we show that free product, which is a noncommutative operation, is associative and respects matroid duality. The free product of matroids M and N is maximal with respect to the weak order among matroids having M as a submatroid, with complementary contraction equal to N. Any minor of the free product of M and N is a free product of a repeated truncation of the corresponding minor of M with a repeated Higgs lift of the corresponding minor of N. We characterize, in terms of their cyclic flats, matroids that are irreducible with respect to free product, and prove that the factorization of a matroid into a free product of irreducibles is unique up to isomorphism. We use these results to determine, for K a field of characteristic zero, the structure of the minor coalgebra of a family of matroids that is closed under formation of minors and free products: namely, is cofree, cogenerated by the set of irreducible matroids belonging to .     

A convergence theorem for spectral factorization

This paper presents a convergence theorem for an iterative method of spectral factorization in the context of multivariate prediction theory. It may be viewed as a constructive proof that the factorization exists, using only the analytic results of Hardy space theory.     

A grothendieck factorization theorem on 2-convex schatten spaces

We prove that for every bounded linear operatorT:C ^2p →H(1≤p<∞,H is a Hilbert space,C ^{2 p} p is the Schatten space) there exists a continuous linear formf onC ^p such thatf≥0, ‖f‖(C _C ^p)*=1 and $$\forall x \in C^{2p} , \left\| {T(x)} \right\| \leqslant 2\sqrt 2 \left\| T \right\|< f\frac{{x * x + xx*}}{2} > 1/2$$ . Forp=∞ this non-commutative analogue of Grothendieck’s theorem was first proved by G. Pisier. In the above statement the Schatten spaceC ^2p can be replaced byE _E ² whereE ⁽²⁾ is the 2-convexification of the symmetric sequence spaceE, andf is a continuous linear form onC _E. The statement can also be extended toL _E{(su2)}(M, τ) whereM is a Von Neumann algebra,τ a trace onM, E a symmetric function space.     

Extension of the Weierstrass factorization theorem

Dans cette note, nous montrons qu’il est possible d’étendre le théorème de factorisation de Weierstrass pour certains éléments d’une algèbre de BanachA.     

A factorization theorem for comaps of geometric lattices

A function γ: K → L between two geometric lattices K and L is a normalized comap if it preserves the relations: x covers or equals y, meets of modular pairs, and the minimum. The theorem, a normalized comap can be factored into an injection followed by a retraction onto a modular flat, is proved.     

A Kronecker factorization theorem for the exceptional Jordan superalgebra

We prove that a Jordan superalgebra J containing the 10-dimensional exceptional Kac superalgebra K₁₀ is isomorphic to (K₁₀⊗_FS)⊕J′, where S is an associative commutative algebra.     

A factorization of the symmetric Pascal matrix involving the Fibonacci matrix

In this short note, we give a factorization of the Pascal matrix. This result was apparently missed by Lee et al. [Some combinatorial identities via Fibonacci numbers, Discrete Appl. Math. 130 (2003) 527-534].