Robustness of optimal designs for correlated random variables

Suppose that Y = (Y_i) is a normal random vector with mean Xb and covariance σ²I_n, where b is a p-dimensional vector (b_j), X = (X_ij) is an n × p matrix with X_ij ∈ {−1, 1}; this corresponds to a factorial design with −1, 1 representing low or high level respectively, or corresponds to a weighing design with −1, 1 representing an object j with weight b_j placed on the left and right of a chemical balance respectively. E-optimal designs Z are chosen that are robust in the sense that they remain E-optimal when the covariance of Y_i, Y_i′ is ρ > 0 for i ≠ i′. Within a smaller class of designs similar results are obtained with respect to a general class of optimality criteria which include the A- and D-criteria.     

On two perturbation estimates of the extreme solutions to the equations X ± A*XA = Q

Two perturbation estimates for maximal positive definite solutions of equations X + A*X⁻¹A = Q and X − A*X⁻¹A = Q are considered. These estimates are proved in [Hasanov et al., Improved perturbation Estimates for the Matrix Equations X ± A*X⁻¹A = Q, Linear Algebra Appl. 379 (2004) 113-135]. We derive new perturbation estimates under weaker restrictions on coefficient matrices of the equations. The theoretical results are illustrated by numerical examples.     

An asymptotic behavior of QR decomposition

The m-th root of the diagonal of the upper triangular matrix R_m in the QR decomposition of AX^mB = Q_mR_m converges and the limit is given by the moduli of the eigenvalues of X with some ordering, where A,B,X∈C_n×n are nonsingular. The asymptotic behavior of the strictly upper triangular part of R_m is discussed. Some computational experiments are discussed.     

Length problems for automorphisms of modules over local rings

Let R be a local ring and M a free module of a finite rank over R. An element τ ∈ Aut_RM is said to be simple if τ ≠ 1 fixes a hyperplane of M.We shall show that for any σ ∈ Aut_RM there exist a basis X for M and ρ ∈ Aut_RM such that ρ acts as a permutation on X and ρ⁻¹σ is a product of m or less than m simple elements in Aut_RM, where m is the order of the invariant factors of σ modulo the maximal ideal of R.Also we shall investigate the problem treated by E.W. Ellers and H. Ishibashi [Factorizations of transformations over a valuation ring, Linear Algebra Appl. 85 (1987) 17-27], in which they showed that σ is a product of simple elements and gave an upper bound of the smallest number of such factors of σ, whereas in the present paper we will give lower bounds of σ in case that R is a local domain. Moreover we will factorize θσ as a product of symmetries and transvections for some θ the matrix of which is diagonal.     

Alexandrov’s inequality and conjectures on some Toeplitz matrices

We study determinant inequalities for certain Toeplitz-like matrices over C. For fixed n and N ? 1, let Q be the n × (n + N − 1) zero-one Toeplitz matrix with Q_ij = 1 for 0 ? j − i ? N − 1 and Q_ij = 0 otherwise. We prove that det(QQ^∗) is the minimum of det(RR^∗) over all complex matrices R with the same dimensions as Q satisfying ∣R_ij∣ ? 1 whenever Q_ij = 1 and R_ij = 0 otherwise. Although R has a Toeplitz-like band structure, it is not required to be actually Toeplitz. Our proof involves Alexandrov’s inequality for polarized determinants and its generalizations. This problem is motivated by Littlewood’s conjecture on the minimum 1-norm of N-term exponential sums on the unit circle. We also discuss polarized Bazin-Reiss-Picquet identities, some connections with k-tree enumeration, and analogous conjectured inequalities for the elementary symmetric functions of QQ^∗.     

k-Potence preserving maps without the linearity and surjectivity assumptions

Let M_n be the space of all n × n complex matrices, and let Γ_n be the subset of M_n consisting of all n × n k-potent matrices. We denote by Ψ_n the set of all maps on M_n satisfying A − λB ∈ Γ_n if and only if ?(A) − λ?(B) ∈ Γ_n for every A,B ∈ M_n and λ ∈ C. It was shown that ? ∈ Ψ_n if and only if there exist an invertible matrix P ∈ M_n and c ∈ C with c^k−1 = 1 such that either ?(A) = cPAP⁻¹ for every A ∈ M_n, or ?(A) = cPA^TP⁻¹ for every A ∈ M_n.     

An alternative approach to unitoidness

In a recent paper [C.R. Johnson, S. Furtado, A generalization of Sylvester’s law of inertia, Linear Algebra Appl. 338 (2001) 287-290], Sylvester’s law of inertia is generalized to any matrix that is ∗-congruent to a diagonal matrix. Such a matrix is called unitoid. In the present paper, an alternative approach to the subject of unitoidness is offered. Specifically, Sylvester’s law of inertia states that a Hermitian n × n matrix of rank r with inertia (p, q, n − r) is ∗-congruent to the direct sum

eⁱ⁰I_p⊕e^iπI_q⊕0I_n-r.     

Classification of small (0, 1) matrices

Denote by A_n the set of square (0, 1) matrices of order n. The set A_n, n ? 8, is partitioned into row/column permutation equivalence classes enabling derivation of various facts by simple counting. For example, the number of regular (0, 1) matrices of order 8 is 10160459763342013440. Let D_n, S_n denote the set of absolute determinant values and Smith normal forms of matrices from A_n. Denote by a_n the smallest integer not in D_n. The sets D₉ and S₉ are obtained; especially, a₉ = 103. The lower bounds for a_n, 10 ? n ? 19 (exceeding the known lower bound a_n ? 2f_n − 1, where f_n is nth Fibonacci number) are obtained. Row/permutation equivalence classes of A_n correspond to bipartite graphs with n black and n white vertices, and so the other applications of the classification are possible.     

On orthogonal systems of matrix algebras

In this work it is shown that certain interesting types of orthogonal system of subalgebras (whose existence cannot be ruled out by the trivial necessary conditions) cannot exist. In particular, it is proved that there is no orthogonal decomposition of M_n(C)⊗M_n(C)≡M_n²(C) into a number of maximal abelian subalgebras and factors isomorphic to M_n(C) in which the number of factors would be 1 or 3.In addition, some new tools are introduced, too: for example, a quantity c(A,B), which measures “how close” the subalgebras A,B⊂M_n(C) are to being orthogonal. It is shown that in the main cases of interest, c(A^′,B^′) - where A^′ and B^′ are the commutants of A and B, respectively - can be determined by c(A,B) and the dimensions of A and B. The corresponding formula is used to find some further obstructions regarding orthogonal systems.     

Decomposable critical tensors

Let λ = (λ₁, … , λ_t) be a partition of m and its conjugate partition. Denote also by λ the irreducible C-character of S_m associated with λ. Let V be a finite dimensional vector space over C.The reach of an element of the symmetry class of tensors V_λ (symmetry class of tensors associated with λ) is defined. The concept of critical element is introduced, as an element whose reach has dimension equal to . It is observed that, in ∧^mV, the notions of critical element and decomposable element coincide. Known results for decomposable elements of ∧^mV are extended to critical elements of V_λ. In particular, for a basis of ⊗^mV induced by a basis of V, generalized Plücker polynomials are constructed in a way that the set of their common roots contains the set of the families of components of decomposable critical elements of V_λ.     

On the span of Hadamard products of vectors

Let A₁, … , A_k be positive semidefinite matrices and B₁, … , B_k arbitrary complex matrices of order n. We show that

span{(A₁x)°(A₂x)°?°(A_kx)|x∈Cⁿ}=range(A₁°A₂°?°A_k)     

Polar decompositions and related classes of operators in spaces ∏κ

Polar decompositions with respect to an indefinite inner product are studied for bounded linear operators acting on a space. Criteria are given for existence of various forms of the polar decompositions, under the conditions that the range of a given operatorX is closed and that zero is not an irregular critical point of the selfadjoint operatorX ^[*]X. Both real and complex spaces are considered. Relevant classes of operators having a selfadjoint (in the sense of the indefinite inner product) square root, or a selfadjoint logarithm, are characterized.The work of this author was partially supported by INdAM-GNCS and MURSTThe work of this author was partially supported by NSF grant DMS-9988579.     

Strong linear preservers of rank reverse permutability on triangular matrices

Let F be a field with ∣F∣ > 2 and T_n(F) be the set of all n × n upper triangular matrices, where n ? 2. Let k ? 2 be a given integer. A k-tuple of matrices A₁, …, A_k ∈ T_n(F) is called rank reverse permutable if rank(A₁ A₂ ? A_k) = rank(A_k A_k−1 ? A₁). We characterize the linear maps on T_n(F) that strongly preserve the set of rank reverse permutable matrix k-tuples.     

Transition matrices for well-conditioned Markov chains

Let T∈R^n×n be an irreducible stochastic matrix with stationary distribution vector π. Set A = I − T, and define the quantity , where A_j, j = 1, … , n, are the (n − 1) × (n − 1) principal submatrices of A obtained by deleting the jth row and column of A. Results of Cho and Meyer, and of Kirkland show that κ₃ provides a sensitive measure of the conditioning of π under perturbation of T. Moreover, it is known that .In this paper, we investigate the class of irreducible stochastic matrices T of order n such that , for such matrices correspond to Markov chains with desirable conditioning properties. We identify some restrictions on the zero-nonzero patterns of such matrices, and construct several infinite classes of matrices for which κ₃ is as small as possible.     

A note on the perturbation of positive matrices by normal and unitary matrices

In a recent paper, Neumann and Sze considered for an n × n nonnegative matrix A, the minimization and maximization of ρ(A + S), the spectral radius of (A + S), as S ranges over all the doubly stochastic matrices. They showed that both extremal values are always attained at an n × n permutation matrix. As a permutation matrix is a particular case of a normal matrix whose spectral radius is 1, we consider here, for positive matrices A such that (A + N) is a nonnegative matrix, for all normal matrices N whose spectral radius is 1, the minimization and maximization problems of ρ(A + N) as N ranges over all such matrices. We show that the extremal values always occur at an n × n real unitary matrix. We compare our results with a less recent work of Han, Neumann, and Tastsomeros in which the maximum value of ρ(A + X) over all n × n real matrices X of Frobenius norm was sought.     

The nullity and rank of linear combinations of idempotent matrices

Baksalary and Baksalary [J.K. Baksalary, O.M. Baksalary, Nonsingularity of linear combinations of idempotent matrices, Linear Algebra Appl. 388 (2004) 25-29] proved that the nonsingularity of P₁ + P₂, where P₁ and P₂ are idempotent matrices, is equivalent to the nonsingularity of any linear combinations c₁P₁ + c₂P₂, where c₁, c₂ ≠ 0 and c₁ + c₂ ≠ 0. In the present note this result is strengthened by showing that the nullity and rank of c₁P₁ + c₂P₂ are constant. Furthermore, a simple proof of the rank formula of Groß and Trenkler [J. Groß, G. Trenkler, Nonsingularity of the difference of two oblique projectors, SIAM J. Matrix Anal. Appl. 21 (1999) 390-395] is obtained.     

Lower bounds for the minimum eigenvalue of Hadamard product of an M-matrix and its inverse

For the Hadamard product A ° A⁻¹ of an M-matrix A and its inverse A⁻¹, we give new lower bounds for the minimum eigenvalue of A ° A⁻¹. These bounds are strong enough to prove the conjecture of Fiedler and Markham [An inequality for the Hadamard product of an M-matrix and inverse M-matrix, Linear Algebra Appl. 101 (1988) 1-8].     

On extremal matrices of second largest exponent by Boolean rank

Let b = b(A) be the Boolean rank of an n × n primitive Boolean matrix A and exp(A) be the exponent of A. Then exp(A) ? (b − 1)² + 2, and the matrices for which equality occurs have been determined in [D.A. Gregory, S.J. Kirkland, N.J. Pullman, A bound on the exponent of a primitive matrix using Boolean rank, Linear Algebra Appl. 217 (1995) 101-116]. In this paper, we show that for each 3 ? b ? n − 1, there are n × n primitive Boolean matrices A with b(A) = b such that exp(A) = (b − 1)² + 1, and we explicitly describe all such matrices.