On the covariance of the Moore-Penrose inverse

Given a square complex matrix A with Moore-Penrose inverse A², we describe the class of invertible matrices T such that (TAT^-1)²=TA²T^-1.     

Seminormality and quasinormality of group rings

We study the permanence of the properties Pic(A) = Pic(A[T]) (seminormality) and Pic(A) = Pic(A[T,T^-1]) (quasinormality) to a finite abelian group ring over the ring A, generalizing results of Bass-Murthy and Pedrini.     

Transposing a matrix by similarity operations

A matrix T is said to co-transpose a square matrix A if T^?1AT=A′ and T^?1A′T=A. For every n?3 there exists a real n×n matrix which cannot be co-transposed by any matrix. However, it is shown that the following classes of real matrices can be co-transposed by a symmetric matrix of order two: 2×2 matrices, normal matrices, and matrices whose square is symmetric.     

A trace inequality for M-matrices and the symmetrizability of a real matrix by a positive diagonal matrix

The question of whether a real matrix is symmetrizable via multiplication by a diagonal matrix with positive diagonal entries is reduced to the corresponding question for M-matrices and related to Hadamard products. In the process, for a nonsingular M-matrix A, it is shown that tr(A^-1A^T) ? n, with equality if and only if A is symmetric, and that the minimum eigenvalue of A^-1 ° A is ? 1 with equality in the irreducible case if and only if A is positive diagonally symmetrizable.     

The Terwilliger algebra of a distance-regular graph of negative type

Let Γ denote a distance-regular graph with diameter D?3. Assume Γ has classical parameters (D,b,α,β) with b<-1. Let X denote the vertex set of Γ and let A∈Mat_X(C) denote the adjacency matrix of Γ. Fix x∈X and let A^∗∈Mat_X(C) denote the corresponding dual adjacency matrix. Let T denote the subalgebra of Mat_X(C) generated by A,A^∗. We call T the Terwilliger algebra of Γ with respect to x. We show that up to isomorphism there exist exactly two irreducible T-modules with endpoint 1; their dimensions are D and 2D-2. For these T-modules we display a basis consisting of eigenvectors for A^∗, and for each basis we give the action of A.     

On Buzyakova's example; quotients of the Cantor trees

Let T be the Cantor tree and let A be a subset of the ωth level of T (= Cantor set C). Buzyakova considered the quotient space TAT^′ obtained from T×2 by identifying two points 〈a,0〉 and 〈a,1〉 for each a∈A to construct an example of a non-submetrizable space of countable extent with a Gδ-diagonal. We prove that the space TAT^′ is submetrizable if and only if C?A is an Fσ-set in C with the Euclidean topology. This improves Buzyakova's Lemma.     

Eine Bemerkung zu einem Satz von W. Hahn aus der Stabilitätstheorie linearer Systeme

A short proof of the following theorem ofW. Hahn is given: LetP be a real n×n matrix,B=1/2(P+P ^T ),A=1/2(P?P ^T ) and letB be negative semidefinite. All eigenvalues ofA have negative real parts if and only if, rank (B, AB, ..., A ^n?1 B)=n.     

On the nonsingular symmetric factors of a real matrix

For a given real square matrix A this paper describes the following matrices: (¹) all nonsingular real symmetric (r.s.) matrices S such that A = S^?1T for some symmetric matrix T.All the signatures (defined as the absolute value of the difference of the number of positive eigenvalues and the number of negative eigenvalues) possible for feasible S in (¹) can be derived from the real Jordan normal form of A. In particular, for any A there is always a nonsingular r.s. matrix S with signature S ? 1 such that A = S^?1T.     

Conjugate cone characterization of positive definite and semidefinite matrices

Positive definite and semidefinite matrices are characterized in terms of positive definiteness and semidefiniteness on arbitrary closed convex cones in Rⁿ. These results are obtained by generalizing Moreau's polar decomposition to a conjugate decomposition. Some typical results are: The matrix A is positive definite if and only if for some closed convex cone K, A is positive definite on K and (A+A^T)^?1 exists and is semidefinite on the polar cone K°. The matrix A is positive semidefinite if and only if for some closed convex cone K such that either K is polyhedral or (A+A^T)(K) is closed, A is positive semidefinite on both K and the conjugate cone K^A={s∣x^T(A+ A^T)s?0?x∈K}, and (A+A^T)x=0 for all x in K such that x^TAx=0.     

An elementary construction of tilting complexes

Let A be an artin algebra and e∈A an idempotent with add(eA_A)=add(D(_AAe)). Then a projective resolution of Ae_eAe gives rise to tilting complexes for A, where P(l)^• is of term length l+1. In particular, if A is self-injective, then is self-injective and has the same Nakayama permutation as A. In case A is a finite dimensional algebra over a field and eAe is a Nakayama algebra, a projective resolution of eAe over the enveloping algebra of eAe gives rise to two-sided tilting complexes {T(2l)^•}_l?1 for A, where T(2l)^• is of term length 2l+1. In particular, if eAe is of Loewy length two, then we get tilting complexes {T(l)^•}_l?1 for A, where T(l)^• is of term length l+1.     

The multiplicity and the Cohen-Macaulayness of extended Rees algebras of equimultiple ideals

Let I be an equimultiple ideal of Noetherian local ring A. This paper gives some multiplicity formulas of the extended Rees algebras T=A[It,t^-1]. In the case A generalized Cohen-Macaulay, we determine when T is Cohen-Macaulay and as an immediate consequence we obtain e.g., some criteria for the Cohen-Macaulayness of Rees algebra R(I) over a Cohen-Macaulay ring in terms of reduction numbers and ideals.     

A quadratic approximation for protein sequence to structure mapping

The existence and representations of some generalized inverses, includingA _{T, *} ⁽²⁾ ,A _{T, *} ^(1,2) ,A _{T, *} ^(2,3) ,A _*,S ⁽²⁾ ,A _*,S ^(1,2) andA _*,S ^(2,4) , are showed. As applications, the perturbation theory for the generalized inverseA _T,S ⁽²⁾ and the perturbation bound for unique solution of the general restricted systemAx=b (dim (AT)=dimT,b∈AT andx∈T) are studied. Moreover, a characterization and representation of the generalized inverseA _{T, *} ^Emphasis>(2) is obtained.     

Addition requirements for matrix and transposed matrix products

Let M be an s × t matrix and let M^T be the transpose of M. Let x and y be t- and s-dimensional indeterminate column vectors, respectively. We show that any linear algorithm A that computes Mx has associated with it a natural dual linear algorithm denoted A^T that computes M^Ty. Furthermore, if M has no zero rows or columns then the number of additions used by A^T exceeds the number of additions used by A by exactly s − t. In addition, a strong correspondence is established between linear algorithms that compute the product Mx and bilinear algorithms that compute the bilinear form y^TMx.     

Max algebraic powers of irreducible matrices in the periodic regime: An application of cyclic classes

In max algebra it is well known that the sequence of max algebraic powers A^k, with A an irreducible square matrix, becomes periodic after a finite transient time T(A), and the ultimate period γ is equal to the cyclicity of the critical graph of A.In this connection, we study computational complexity of the following problems: (1) for a given k, compute a periodic power A^r with and r?T(A), (2) for a given x, find the ultimate period of {A^l⊗x}. We show that both problems can be solved by matrix squaring in O(n³logn) operations. The main idea is to apply an appropriate diagonal similarity scaling A?X^-1AX, called visualization scaling, and to study the role of cyclic classes of the critical graph.     

On the characterization of generalized inverses by bordered matrices

A method to characterize the class of all generalized inverses of any given matrix A is considered. Given a matrix A and a nonsingular bordered matrix T of A,

$\text{T=}\text{A}\text{P}\text{Q}\text{R}$

the submatrix, corresponding to A, of T^-1 is a generalized inverse of A, and conversely, any generalized inverse of A is obtainable by this method. There are different definitions of a generalized inverse, and the arguments are developed with the least restrictive definition. The characterization of the Moore-Penrose inverse, the most restrictive definition, is also considered.     

Power symmetric stochastic matrices

Stochastic matrices A which satisfy the equation A^T=A^p are characterized for integral values of p > 1.     

Bohr compactifications and a result of følner

In this paper, we study the Bohr compactification of an arbitrary topological groupT with regard to obtaining relations between relatively dense (or discretely syndetic) subsets ofT, and neighborhoods of the identity in the Bohr compactification. The methods utilized are those algebraic techniques which have been recently applied to topological dynamics (see [2]). For an abelian group, we show that cls (A ^?1 AAa ^?1), forA relatively dense anda∈A, is usually a neighborhood of the identity, thus generalizing a result of Følner [4]. Moreover, an analogous result is proved in the non-abelian case under additional assumptions. Finally, we utilize these results to obtain a generalization of a result of Cotlar-Ricabarra [1] concerning maximal almost periodicity in abelian topological groups.     

On Lebesgue Measure of Integral Self-Affine Sets

Let A be an expanding integer n×n matrix and D be a finite subset of ? ⁿ . The self-affine set T=T(A,D) is the unique compact set satisfying the equality \(A(T)=\bigcup_{d\in D}(T+d)\). We present an effective algorithm to compute the Lebesgue measure of the self-affine set T, the measure of the intersection T∩(T+u) for u∈? ⁿ , and the measure of the intersection of self-affine sets T(A,D ₁)∩T(A,D ₂) for different sets D ₁, D ₂?? ⁿ .     

Characterization of Hermitian and skew-Hermitian maps between matrix algebras

Let D be a division ring with an involution J such that D is finite-dimensional over its center Z and char D≠2. Let T:M_m(D)→M_n(D) be a Z-linear map between matrix rings over D. We show that T satisfies [T(X)]¹=T(X¹) if and only if T(X)=∑±A¹_kXA_k. Similarly, T satisfies [T(X)]¹ = ? T(X¹) if and only if T(X = ∑(A¹_kXB_k ? B¹_kXA_k). The first of these results generalizes and extends a theorem of R.D. Hill [2] on Hermitian-preserving transformations.