Canonical forms for linear systems

This paper considers canonical forms for the similarity action of Gl(n) on $\text{∑}{}_{\mathrm{n,m}}\text{={(A,B)∈}\text{C}{}^{\mathrm{n}}\text{·}{}^{\mathrm{n}}\text{×}\text{C}{}^{\mathrm{n}}\text{·}{}^{\mathrm{m}}\text{}}$:

$\text{Gl(n×∑}{}_{\mathrm{n,m}}\text{→∑}{}_{\mathrm{n,m}}$

,

$\text{(H,(A,B))?(HAH}{}^{-1}\text{,HB)}$

Those canonical forms are obtained as an application of a more general method to select canonical elements M_c in the orbits $\text{O}{}_{\mathrm{M}}$ of a matrix group G acting on a set of matrices $\text{M}\text{?}\text{C}{}^{\mathrm{l}}\text{·}{}^{\mathrm{p}}$. We define a total order (?) on $\text{C}{}^{\mathrm{l}}\text{·}{}^{\mathrm{p}}$, different from the lexicographic order ^l? [0^l?x ? x <0, but $\text{0?x≠0 for x∈}\text{R}\text{]}$ and consider normalized $\text{O}{}_{\mathrm{M}}$-elements with a minimal number of parameters:

$\text{min{}\text{M}\text{?}\text{∈}\text{O}{}_{\mathrm{M}}\text{:}\text{M}\text{?}\text{normalized}}$

It is shown that the row and column echelon forms, the Jordan canonical form, and “nice” control canonical forms for reachable (A,B)-pairs have a homogeneous interpretation as such (?)-minimal orbit elements. Moreover new canonical forms for the general action (?) are determined via this method.     

Factored forms for solutions of AX − XB = C and X − AXB = C in companion matrices

The main concern of this paper is linear matrix equations with block-companion matrix coefficients. It is shown that general matrix equations AX ? XB = C and X ? AXB = C can be transformed to equations whose coefficients are block companion matrices: $\text{C}\text{?}{}_{\mathrm{L}}\text{X?XC}{}_{\mathrm{M}}\text{=}\text{diag}\text{[I 0…0]}$ and $\text{X?}\text{C}\text{?}{}_{\mathrm{L}}\text{XC}{}_{\mathrm{M}}\text{=}\text{diag}\text{[I 0…0]}$, respectively, where ?_L and C_M stand for the first and second block-companion matrices of some monic r × r matrix polynomials L(λ) = λ^sI + Σ^s?1_j=0λ^jL_j and M(λ) = λ^tI + Σ^t7minus;1_j=0λ^jM_j. The solution of the equat with block companion coefficients is reduced to solving vector equations Sx = ?, where the matrix S is r²l × r²l[l = max(s, t)] and enjoys some symmetry properties.     

On the completely positive and positive-semidefinite-preserving cones

The cone CP_n,q of completely positive linear transformations from $\text{M}{}_{\mathrm{n}}\text{(}\text{C}\text{)=}\text{M}{}_{\mathrm{n}}$ to $\text{M}$_q is shown to be isometrically isomorphic to $\text{P}$_nq, the cone of nq by nq positive semidefinite matrices. Generalizations of scalar and matrix results to $\text{CP}{}_{\mathrm{n,\; q}}\text{?}\text{HP}{}_{\mathrm{n,\; q}}\text{?}\text{L}\text{(}\text{M}{}_{\mathrm{n}}\text{,}\text{M}{}_{\mathrm{q}}\text{)}$ (where HP_n,q represents the hermitian-preserving linear transformations) are discussed. Relationships among the completely positives, the set of positive semidefinite preservers $\text{π(}\text{P}{}_{\mathrm{n}}\text{)}$, and its dual $\text{π(}\text{M}{}_{\mathrm{n}}\text{)}{}^1$ are given. Left and right facial ideals of CP are characterized. Properties of the joint angular field of values of a finite sequence of hermitian matrices H₁,…, H_m are studied, leading to a characterization of $\text{π(}\text{P}{}_{\mathrm{q}}\text{,}\text{P}{}_{\mathrm{n}}\text{)}$.     

Parallel calculation of a linear mapping on a computer network

We are interested in the parallel computation of a linear mapping of n real variables by a network of computers with restricted means of communication between them and without any common memory. Let $\text{M}{}_{\mathrm{n\times n}}\text{(}\text{R}\text{)}$ denote the algebra of n×n real matrices, and let G be the graph associated with a binary, reflexive and symmetric relation R over {1,2, …,n}. We define

$\text{A}{}_{\mathrm{R}}\text{= {A?M}{}_{\mathrm{n\times n}}\text{(}\text{R}\text{):a}{}_{\mathrm{ij}}\text{≠ 0}\text{implies}\text{iRj}}$

A matrix $\text{M∈M}{}_{\mathrm{n\times n}}\text{(}\text{R}\text{)}$ is said to be realizable on G if it can be expressed as a product of elements of A_R. Therefore, every matrix of $\text{M}{}_{\mathrm{n\times n}}\text{(}\text{R}\text{)}$ is realizable on G if and only if A_R generates $\text{M}{}_{\mathrm{n\times n}}\text{(}\text{R}\text{)}$. We show that A_R generates $\text{M}{}_{\mathrm{n\times n}}\text{(}\text{R}\text{)}$ if and only if G is connected.     

Spaces of matrices of equal rank

Let M_m,n(F) denote the space of all mXn matrices over the algebraically closed field F. A subspace of M_m,n(F), all of whose nonzero elements have rank k, is said to be essentially decomposable if there exist nonsingular mXn matrices U and V respectively such that for any element A, UAV has the form

$\text{UAV=}\text{A}{}_1\text{A}{}_2\text{A}{}_3\text{0}$

where A₁ is iX(k–i) for some i?k. Theorem: If $\text{K}$ is a space of rank k matrices, then either $\text{K}$ is essentially decomposable or dim $\text{K}$?k+1. An example shows that the above bound on non-essentially-decomposable spaces of rank k matrices is sharp whenever n?2k–1.     

Completely positive linear maps on complex matrices

A linear map Φ from $\text{M}$_n to $\text{M}$_m is completely positive iff it admits an expression $\text{Φ(A)=Σ}{}_{\mathrm{i}}\text{V}{}^1{}_{\mathrm{i}}\text{AV}{}_{\mathrm{i}}$ where V_i are n×m matrices.     

A fundamental theorem on lambda-matrices with applications —II. Difference equations with constant coefficients

The fundamental theorem of the title refers to a spectral resolution for the inverse of a lambda-matrix $\text{L(λ) =}\text{∑}\text{i=0}\text{l}\text{A}{}_{\mathrm{i}}\text{λ}{}^{\mathrm{i}}$ where the A_i are n×n complex matrices and detA_l ≠ 0. In this paper general solutions are formulated for difference equations of the form $\text{∑}\text{i=0}\text{l}\text{A}{}_{\mathrm{i}}\text{u}{}_{\mathrm{r\; +\; i}}\text{= ?}{}_{\mathrm{\gamma }}\text{, r = 1, 2,…}$. The use of these solutions is illustrated i new proof of Franklin's results describing the sums of powers of the eigenvalues of L(λ) (the generalized Newton identities), and in obtaining convergence proofs for the application of Bernoulli's method to the solution of $\text{∑}\text{i=0}\text{l}\text{A}{}_{\mathrm{i}}\text{S}{}^{\mathrm{i}}\text{= 0}$ for matrix S.     

Weak convergence of compound stochastic process,I

Compound stochastic processes are constructed by taking the superpositive of independent copies of secondary processes, each of which is initiated at an epoch of a renewal process called the primary process. Suppose there are M possible k-dimensional secondary processes {ξ^v(t):t?0}, v=1,2,…,M. At each epoch of the renewal process {A(t):t?0} we initiate a random number of each of the M types. Let m_l:l?1} be a sequence of M-dimensional random vectors whose components specify the number of secondary processes of each type initiated at the various epochs. The compound process we study is

, where the ξ^v_lj() are independent copies of ξ^v,m_lv is the vth component of m and {τ_l:l?1} are the epochs of the renewal process. Our interest in this paper is to obtain functional central limit theorems for {Y(t):t?0} after appropriately scaling the time parameter and state space. A variety of applications are discussed.     

Higher analogues of Stickelberger's theorem

Let l be an odd prime number, F denote any totally real number field and E/F be an Abelian CM extension of F of conductor f. In this paper we prove that for every n odd and almost all prime numbers l we have where S_n(E/F,l) is the Stickelberger ideal (Ann. of Math. 135 (1992) 325–360; J. Coates, p-adic L-functions and Iwasawa's theory, in: Algebraic Number Fields by A. Fröhlich, Academic Press, London, 1977). In addition if we assume the Quillen–Lichtenbaum conjecture then To cite this article: G. Banaszak, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

A nonmetrizable manifold from ♢+

Assuming ♢⁺, a perfectly normal 3-dimensional manifold M is constructed with the property that M=∪_α<ω₁M_α where each M_α is an open connected metric subspace of M with $\text{∪}{}_{\mathrm{\beta <\alpha }}\text{M}{}_{\mathrm{\beta }}\text{⫋M}{}_{\mathrm{\alpha }}$.     

Generalized Schur complements

Let A be an n×n complex matrix. For a suitable subspace $\text{M}$ of Cⁿ the Schur compression A _$\text{M}$ and the (generalized) Schur complement A_/$\text{M}$ are defined. If A is written in the form

$\text{A=}\text{B}\text{C}\text{S}\text{T}$

according to the decomposition $\text{C}{}^{\mathrm{n}}\text{=}\text{M}\text{⊕}\text{M}{}^{\mathrm{\perp }}$ and if B is invertible, then

$\text{A}{}_{\mathrm{<mtext>M</mtext>}}\text{=}\text{B}\text{C}\text{S}\text{SB}{}^{\mathrm{?1}}\text{C}$

and

$\text{A}{}_{\mathrm{<mtext>/</mtext><mtext>M</mtext>}}\text{=}\text{0}\text{0}\text{0}\text{T?SB}{}^{\mathrm{?1}}\text{C}\text{·}$

The commutativity rule for Schur complements is proved:

$\text{(A}{}_{\mathrm{<mtext>/</mtext><mtext>M</mtext>}}\text{)}{}_{\mathrm{<mtext>/</mtext><mtext>N</mtext>}}\text{=(A)}{}_{\mathrm{<mtext>/</mtext><mtext>N</mtext>}}\text{)}{}_{\mathrm{<mtext>/</mtext><mtext>M</mtext>}}\text{·}$

This unifies Crabtree and Haynsworth's quotient formula for (classical) Schur complements and Anderson's commutativity rule for shorted operators. Further, the absorption rule for Schur compressions is proved:

$\text{(A}{}_{\mathrm{<mtext>/</mtext><mtext>M</mtext>}}\text{)}{}_{\mathrm{<mtext>N</mtext>}}\text{=(A}{}_{\mathrm{<mtext>N</mtext>}}\text{)}{}_{\mathrm{<mtext>M</mtext>}}\text{=A}{}_{\mathrm{<mtext>M</mtext>}}\text{whenever}\text{M}\text{?}\text{N}$

.     

Duality in hypercomplex function theory

Let $\text{A}$ be the Clifford algebra constructed over a quadratic n-dimensional real vector space with orthogonal basis {e₁,…, e_n}, and e₀ be the identity of $\text{A}$. Furthermore, let M_k(Ω;$\text{A}$) be the set of $\text{A}$-valued functions defined in an open subset Ω of R^m+1 (1 ? m ? n) which satisfy D^kf = 0 in Ω, where D is the generalized Cauchy-Riemann operator $\text{D = ∑}{}_{\mathrm{i\; =\; 0}}{}^{\mathrm{m}}\text{e}{}_{\mathrm{i}}\text{(}\text{?}\text{?x}{}_{\mathrm{i}}\text{)}$ and k? N. The aim of this paper is to characterize the dual and bidual of M_k(Ω;$\text{A}$). It is proved that, if M_k(Ω;$\text{A}$) is provided with the topology of uniform compact convergence, then its strong dual is topologically isomorphic to an inductive limit space of Fréchet modules, which in its turn admits M_k(Ω;$\text{A}$) as its dual. In this way, classical results about the spaces of holomorphic functions and analytic functionals are generalized.     

On tensor products of certain group C1-algebras

If A and B are C¹-algebras there is, in general, a multiplicity of C¹-norms on their algebraic tensor product A ⊙ B, including maximal and minimal norms ν and α, respectively. A is said to be nuclear if α and ν coincide, for arbitrary B. The earliest example, due to Takesaki [11], of a nonnuclear C¹-algebra was $\text{C}{}_{\mathrm{l}}{}^1\text{(F}{}_2\text{)}$, the C¹-algebra generated by the left regular representation of the free group on two generators F₂. It is shown here that W¹-algebras, with the exception of certain finite type I's, are nonnuclear.If $\text{C}{}^1\text{(F}{}_2\text{)}$ is the group C¹-algebra of F₂, there is a canonical homomorphism λ_l of $\text{C}{}^1\text{(F}{}_2\text{)}$ onto $\text{C}{}_{\mathrm{l}}{}^1\text{(F}{}_2\text{)}$. The principal result of this paper is that there is a norm ζ on $\text{C}{}_{\mathrm{l}}{}^1\text{(F}{}_2\text{) ⊙ C}{}_{\mathrm{l}}{}^1\text{(F}{}_2\text{)}$, distinct from α, relative to which the homomorphism $\text{λ ⊙ λ}{}_{\mathrm{l}}\text{: C}{}^1\text{(F}{}_2\text{) ⊙ C}{}^1\text{(F}{}_2\text{) → C}{}_{\mathrm{l}}{}^1\text{(F}{}_2\text{) ⊙ C}{}_{\mathrm{l}}{}^1\text{(F}{}_2\text{)}$ is bounded ($\text{C}{}^1\text{(F}{}_2\text{) ⊙ C}{}^1\text{(F}{}_2\text{)}$ being endowed with the norm α). Thus quotients do not, in general, respect the norm α; a consequence of this is that the set of ideals of the α-tensor product of C¹-algebras A and B may properly contain the set of product ideals {$\text{I ? B + A ? J: I ? A, J ? B}$}.Let A and B be C¹-algebras. If A or B is a W¹-algebra there are on A ⊙ B certain C¹-norms, defined recently by Effros and Lance [3], the definitions of which take account of normality. In the final section of the paper it is shown by example that these norms, with α and ν, can be mutually distinct.     

A new approach to the Tomita-Takesaki theory of generalized Hilbert algebras

Let M be a von Neumann algebra with separating and cyclic vector ξ₀. The map $\text{xξ}{}_0\text{→ x}{}^1\text{ξ}{}_0$ with x?M has a least closed extension S. Tomita proved that the isometric involution J and the positive self-adjoint operator Δ obtained from the polar decomposition $\text{S = JΔ}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ of S satisfy JMJ = M′ and Δ^itMΔ^?it = M for any real t. More generally, he obtained similar results for the left von Neumann algebra of any generalized Hilbert algebra. In this paper a shorter proof of his results is given.     

Injectivity and operator spaces

Let B(H) be the bounded operators on a Hilbert space H. A linear subspace R ? B(H) is said to be an operator system if 1 ?R and R is self-adjoint. Consider the category $\text{b}$ of operator systems and completely positive linear maps. R ∈ $\text{C}$ is said to be injective if given A ? B, A, B ∈ $\text{C}$, each map A → R extends to B. Then each injective operator system is isomorphic to a conditionally complete C¹-algebra. Injective von Neumann algebras R are characterized by any one of the following: (1) a relative interpolation property, (2) a finite “projectivity” property, (3) letting M_m = B(C^m), each map R → N ? M_m has approximate factorizations R → M_n → N, (4) letting K be the orthogonal complement of an operator system N ? M_m, each map $\text{M}{}_{\mathrm{m}}\text{K}\text{→ R}$ has approximate factorizations $\text{M}{}_{\mathrm{m}}\text{K}\text{→ M}{}_{\mathrm{n}}\text{→ R}$. Analogous characterizations are found for certain classes of C¹-algebras.