Roth's similarity theorem and rank minimization in the presence of nonderogatory or semisimple eigenvalues

Roth's similarity theorem on the consistency of Sylvester's matrix equation AX???XA?=?C can be extended to a theorem on rank minimization if the common eigenvalues of A and B are nonderogatory or semisimple.     

Approximate multiplication of tensor matrices based on the individual filtering of factors

Algorithms are proposed for the approximate calculation of the matrix product $ \tilde C $ \tilde C ≈ C = A · B, where the matrices A and B are given by their tensor decompositions in either canonical or Tucker format of rank r. The matrix C is not calculated as a full array; instead, it is first represented by a similar decomposition with a redundant rank and is then reapproximated (compressed) within the prescribed accuracy to reduce the rank. The available reapproximation algorithms as applied to the above problem require that an array containing r ^2d elements be stored, where d is the dimension of the corresponding space. Due to the memory and speed limitations, these algorithms are inapplicable even for the typical values d = 3 and r ∼ 30. In this paper, methods are proposed that approximate the mode factors of C using individually chosen accuracy criteria. As an application, the three-dimensional Coulomb potential is calculated. It is shown that the proposed methods are efficient if r can be as large as several hundreds and the reapproximation (compression) of C has low complexity compared to the preliminary calculation of the factors in the tensor decomposition of C with a redundant rank.     

Sufficient conditions for permutation equivalence to a WHS-matrix

Square matrices with positive leading principal minors, called WHS-matrices (weak Hawkins–Simon), are considered in economics. Some sufficient conditions for a matrix to be a WHS-matrix after suitable row and/or column permutations have recently appeared in the literature. New and unified proofs and generalizations of some results to rectangular matrices are given. In particular, it is shown that if left multiplication of a rectangular matrix A by some nonnegative matrix is upper triangular with positive diagonal, then some row pemutation of A is a WHS-matrix. For a nonsingular A with either the first nonzero entry of each of its rows positive or the last nonzero entry of each column of A ^?1 positive, again some row permutation of A is a WHS-matrix. In addition, any rectangular full rank semipositive matrix is shown to be permutation equivalent to a WHS-matrix.     

The group reduction for bounded cosine functions on UMD spaces

It is shown that if A generates a bounded cosine operator function on a UMD space X, then i(−A)^1/2 generates a bounded C ₀-group. The proof uses a transference principle for cosine functions.      

C*-Nuclearity implies CPAP

In this paper we show: A C^*-algebra A with identity e is C^*-nuclear (i.e. “nuclear” in the sense of LANCE [3, p. 159] or “property T” in the sense of TAKESAKI [6] if and only if the identity operator id_A on A can be approximated in the strong operator topology by completely positive linear operators V from A into A of finite rank with norm one and V(e)=e. From which follows that every C^*-nuclear C^*-algebra (possible without identity) possesses the CPAP of LANCE [3, def. 3.5.]. This answers questions of LANCE [3, p. 173] and SAKAI [5, p. 64].     

Complete positivity,tensor products and C*-nuclearity for inverse limits of C*-algebras

The paper aims at developing a theory of nuclear (in the topological algebraic sense) pro-C^*-algebras (which are inverse limits of C^*-algebras) by investigating completely positive maps and tensor products. By using the structure of matrix algebras over a pro-C^*-algebra, it is shown that a unital continuous linear map between pro-C^*-algebrasA andB is completely positive iff by restriction, it defines a completely positive map between the C^*-algebrasb(A) andb(B) consisting of all bounded elements ofA andB. In the metrizable case,A andB are homeomorphically isomorphic iff they are matricially order isomorphic. The injective pro-C^*-topology α and the projective pro-C^*-topology v on A⊗B are shown to be minimal and maximal pro-C^*-topologies; and α coincides with the topology of biequicontinous convergence iff eitherA orB is abelian. A nuclear pro-C^*-algebraA is one that satisfies, for any pro-C^*-algebra (or a C^*-algebra)B, any of the equivalent requirements; (i) α =v onA ⊗B (ii)A is inverse limit of nuclear C^*-algebras (iii) there is only one admissible pro-C^*-topologyon A⊗B (iv) the bounded partb(A) ofA is a nuclear C⊗-algebra (v) any continuous complete state map A→B^* can be approximated in simple weak^* convergence by certain finite rank complete state maps. This is used to investigate permanence properties of nuclear pro-C^*-algebras pertaining to subalgebras, quotients and projective and inductive limits. A nuclearity criterion for multiplier algebras (in particular, the multiplier algebra of Pedersen ideal of a C^*-algebra) is developed and the connection of this C^*-algebraic nuclearity with Grothendieck’s linear topological nuclearity is examined. A σ-C^*-algebraA is a nuclear space iff it is an inverse limit of finite dimensional C^*-algebras; and if abelian, thenA is isomorphic to the algebra (pointwise operations) of all scalar sequences.     

Applications of the growth characteristics induced by the spectral distance

Let A be a complex unital Banach algebra. Using a connection between the spectral distance and the growth characteristics of a certain entire map into A, we derive a generalization of Gelfand’s famous power boundedness theorem. Elaborating on these ideas, with the help of a Phragm´en-Lindel¨of device for subharmonic functions, it is then shown, as the main result, that two normal elements of a C^?-algebra are equal if and only if they are quasinilpotent equivalent.     

Diagonalization of compact operators on Hilbert modules overC *-algebras of real rank zero

The classical Hilbert-Schmidt theorem can be extended to compact operators on HilbertA-modules overW ^*-algebras of finite type; i.e., with minor restrictions, compact operators onH* _A can be diagonalized overA. We show that ifB is a weakly denseC ^*-subalgebra ofA with real rank zero and if some additional condition holds, then the natural extension fromH _B toH* _A ⊃H _B of a compact operator can be diagonalized so that the diagonal elements belong to the originalC ^*-algebraB. Translated fromMatematicheskie Zametki, Vol. 62, No. 6, pp. 865–870, December, 1997. Translated by O. V. Sipacheva     

Extended integer rank reduction formulas and Smith normal form

We present an integer rank reduction formula for transforming the rows and columns of an integer matrix A. By repeatedly applying the formula to reduce rank, an extended integer rank reducing process is derived. The process provides a general finite iterative approach for constructing factorizations of A and A ^T under a common framework of a general decomposition V ^T AP?=?Ω. Then, we develop the integer Wedderburn rank reduction formula and its integer biconjugation process. Both the integer biconjugation process associated with the Wedderburn rank reduction process and the scaled extended integer Abaffy–Broyden–Spedicato (ABS) class of algorithms are shown to be in the integer rank reducing process. We also show that the integer biconjugation process can be derived from the scaled integer ABS class of algorithms applied to A or A ^T . Finally, we show that the integer biconjuagation process is a special case of our proposed ABS class of algorithms for computing the Smith normal form.     

Possible numbers of ones in the squares of 0–1 matrices with a given rank

For a symmetric 0–1 matrix A, we give the number of ones in A ² when rank(A) = 1, 2, and give the maximal number of ones in A ² when rank(A) = k (3 ≤ k ≤ n). The sufficient and necessary condition under which the maximal number is achieved is also obtained. For generic 0–1 matrices, we only study the cases of rank 1 and rank 2.     

The spectral theory of commutative C∗-algebras: The constructive Gelfand-Mazur theorem

It is shown, for a commutative C^?-algebra in any Grothendieck topos E, that the locale MFn A of multiplicative linear functionals on A is isomorphic to the locale Max A of maximal ideals of A, extending the classical result that the space of C^?-algebra homomorphisms from A to the field of complex numbers is isomorphic to the maximal ideal space of A, that is, the Gelfand-Mazur theorem, to the constructive context of any Grothendieck topos. The technique is to present Max A, in analogy with our earlier definition of MFn A, by means of a propositional theory which expresses one's natural intuition of the notion involved, and then to establish various properties, leading up to the final result, by formal reasoning within these theories.     

On the tournament matrices with smallest rank

A tournament matrix is a square zero-one matrix A satisfying the equation A+A^t = J ? I, where J is the all-ones matrix. In [1] it was proved that if A is an n × n tournament matrix, then the rank of A is at least (n - 1)/2, over any field; and in characteristic zero rank (A) equals n - 1 or n. Michael [3] has constructed examples having rank (n - 1)/2; they are double borderings of Hadamard tournaments of order n - 2, and so must satisfy n ≡ 1 (mod 4). In this note, we supplement this result by showing that an analogous construction is sometimes impossible when n ≡ 3 (mod 4).     

Equalities for orthogonal projectors and their operations

A complex square matrix A is called an orthogonal projector if A ² = A = A*, where A* denotes the conjugate transpose of A. In this paper, we give a comprehensive investigation to matrix expressions consisting of orthogonal projectors and their properties through ranks of matrices. We first collect some well-known rank formulas for orthogonal projectors and their operations, and then establish various new rank formulas for matrix expressions composed by orthogonal projectors. As applications, we derive necessary and sufficient conditions for various equalities for orthogonal projectors and their operations to hold.     

On topologically finite-dimensional simple <Emphasis Type="Italic">C</Emphasis>*-algebras

We show that, if a simple C*-algebra A is topologically finite-dimensional in a suitable sense, then not only K₀(A) has certain good properties, but A is even accessible to Elliott’s classification program. More precisely, we prove the following results:If A is simple, separable and unital with finite decomposition rank and real rank zero, then K₀(A) is weakly unperforated.If A has finite decomposition rank, real rank zero and the space of extremal tracial states is compact and zero-dimensional, then A has stable rank one and tracial rank zero. As a consequence, if B is another such algebra, and if A and B have isomorphic Elliott invariants and satisfy the Universal coefficients theorem, then they are isomorphic.In the case where A has finite decomposition rank and the space of extremal tracial states is compact and zero-dimensional, we also give a criterion (in terms of the ordered K₀-group) for A to have real rank zero. As a byproduct, we show that there are examples of simple, stably finite and quasidiagonal C*-algebras with infinite decomposition rank.Supported by: EU-Network Quantum Spaces - Noncommutative Geometry (Contract No. HPRN-CT-2002-00280) and Deutsche Forschungsgemeinschaft (SFB 478).     

On principal submatrices

It is shown that if A[ω] is a principal submatrix of the positive definite Hermitian matrix A, then A ^?1[ω] ?(A[ω])^?1is a positive semidefinite hermitian matrix. This fact is used to give a brief proof of a result of Saburou Saitoh concerning Hadamard products.     

Quadratically normal and congruence-normal matrices

A matrix A ∈ C ^n×n is unitarily quasidiagonalizable if A can be brought by a unitary similarity transformation to a block diagonal form with 1 × 1 and 2 × 2 diagonal blocks. In particular, the square roots of normal matrices, i.e., the so-called quadratically normal matrices are unitarily quasidiagonalizable. A matrix A ∈ C ^n×n is congruence-normal if B = A[`(A)] B = A\overline A is a conventional normal matrix. We show that every congruence-normal matrix A can be brought by a unitary congruence transformation to a block diagonal form with 1 × 1 and 2 × 2 diagonal blocks. Our proof emphasizes andexploitsalikenessbetween theequations X <sup>2</sup> = B and X[`(X)] = B X\overline X = B for a normal matrix B. Bibliography: 13 titles.     

Existence of a low rank or ℋ︁‐matrix approximant to the solution of a Sylvester equation

We consider the Sylvester equation AX?XB+C=0 where the matrix C∈?^n×m is of low rank and the spectra of A∈?^n×n and B∈?^m×m are separated by a line. We prove that the singular values of the solution X decay exponentially, that means for any ε∈(0,1) there exists a matrix X? of rank k=O(log(1/ε)) such that ∥X?X?∥₂?ε∥X∥₂. As a generalization we prove that if A,B,C are hierarchical matrices then the solution X can be approximated by the hierarchical matrix format described in Hackbusch (Computing 2000; 62 : 89–108). The blockwise rank of the approximation is again proportional to log(1/ε). Copyright © 2004 John Wiley & Sons, Ltd.     

Primes in the semigroup of non-negative matrices

A matrix A in the semigroup N _n of non-negative n×nmatrices is prime if A is not monomial and A=BC,B CεN _n implies that either B or C is monomial. One necessary and another sufficient condition are given for a matrix in N _n to be prime. It is proved that every prime in N _n is completely decomposable.     

Continuity of The Drazin inverse

Let A be an n×n matrix. It is shown that if a matrix  comes close to satisfying the definition of the Drazin inverse of A,A^D , then  is close to A^D .     

Some results on symplectic matrices

The principal results are that if A is an integral matrix such that AA^T is symplectic then A = CQ, where Q is a permutation matrix and C is symplectic; and that if A is a hermitian positive definite matrix which is symplectic, and B is the unique hermitian positive definite pth.root of A, where p is a positive integer, then B is also symplectic.