The robertson-taussky inequality revisited

Let a positive definite Hermitian matrix HεMⁿ(C) be decomposed as H=A + iB, with A, B ε M_nm(R). We give two new proofs of the inequality det H ≤ det A (with equality iff B = 0. each of which vields something futher. One exhibits majorization between the eigenvalues of A and H the other allows proof of the permanental analog per H≥per A.     

On application of the Lanczos method to solution of some partial differential equations

Let A be a square symmetric n × n matrix, φ be a vector from ⁿ, and f be a function defined on the spectral interval of A. The problem of computation of the vector u = f(A)φ arises very often in mathematical physics.

We propose the following method to compute u. First, perform m steps of the Lanczos method with A and φ. Define the spectral Lanczos decomposition method (SLDM) solution as u_m = φ Qf(H)e₁, where Q is the n × m matrix of the m Lanczos vectors and H is the m × m tridiagonal symmetric matrix of the Lanczos method. We obtain estimates for u − u_m that are stable in the presence of computer round-off errors when using the simple Lanczos method.

We concentrate on computation of exp(− tA)φ, when A is nonnegative definite. Error estimates for this special case show superconvergence of the SLDM solution. Sample computational results are given for the two-dimensional equation of heat conduction. These results show that computational costs are reduced by a factor between 3 and 90 compared to the most efficient explicit time-stepping schemes. Finally, we consider application of SLDM to hyperbolic and elliptic equations.     

A note on linearization of actions of finitely semisimple Hopf algebras on local algebras

Let H be a Hopf algebra over a field k and let H A → A, h a → h.a, be an action of H on a commutative local Noetherian kalgebra (A, m). We say that this action is linearizable if there exists a minimal system x₁, …, x_n of generators of the maximal ideal m such that h.x_i ε kx₁ + …+ kx_n for all h ε H and i = 1, …, n. In the paper we prove that the actions from a certain class are linearizable (see Theorem 4), and we indicate some consequences of this fact.     

The growth of laguerre matrix polynomials on bounded intervals

Let A be a matrix in ^r×r such that Re(z) > −1/2 for all the eigenvalues of A and let {π_n^(A,1/2) (x)} be the normalized sequence of Laguerre matrix polynomials associated with A. In this paper, it is proved that π_n^(A,1/2) (x) = O(n^(A)/2ln^r−1(n)) and π_n+1^(A,1/2) (x) − π_n^(A,1/2) (x) = O(n^((A)−1)/2ln^r−1(n)) uniformly on bounded intervals, where (A) = max{Re(z); z eigenvalue of A}.     

An Extension of an Inequality Involving Symmetric Products with an Application to Permanents

Let A be an nk × nk positive semi-definite symmetric matrix partitioned into blocks A^ij each of which is an n × n matrix. In [2] Mine states a conjecture of Marcus that per(A) ≥ per(G) where G is the k × k matrix [per(A^ij)]. In this paper we prove a weaker inequality namely that per(A) ≥ (k!)^-1per(G).     

Spherical functions and immanants

Let λ be an irreducible character of S_n corresponding to the partition (r,s) of n. Let A be a positive semidefinite Hermitian n × n matrix. Let d_λ(A) and per(A) be the immanants corresponding to λ and to the trivial character of S_n, respectively. A proof of the inequality dλ(A)≤λ(id)per(A) is given.     

A formula for the number of Latin squares

We establish an explicit formula for the number of Latin squares of order n:

, where B_n is the set of n×n(0,1) matrices, σ₀(A is the number of zero elements of the matrix A and per A is the permanent of the matrix A.     

A bound for the condition of a hyperbolic eigenvector matrix

The hyperbolic eigenvector matrix is a matrix X which simultaneously diagonalizes the pair (H,J), where H is Hermitian positive definite and J = diag(±1) such that X*HX = Δ and X*JX = J. We prove that the spectral condition of X, κ(X), is bounded byK(X)√minK(D^*HD), where the minimum is taken over all non-singular matrices D which commute with J. This bound is attainable and it can be simply computed. Similar results hold for other signature matrices J, like in the discretized Klein—Gordon equation.     

Krylov subspace methods for eigenvalues with special properties and their analysis for normal matrices

In this paper we propose a general approach by which eigenvalues with a special property of a given matrix A can be obtained. In this approach we first determine a scalar function ψ: C → C whose modulus is maximized by the eigenvalues that have the special property. Next, we compute the generalized power iterations uinj + 1 = ψ(A)u_j, j = 0, 1,…, where u₀ is an arbitrary initial vector. Finally, we apply known Krylov subspace methods, such as the Arnoldi and Lanczos methods, to the vector u_n for some sufficiently large n. We can also apply the simultaneous iteration method to the subspace span{x⁽ⁿ⁾₁,…,x⁽ⁿ⁾_k} with some sufficiently large n, where x^(j+1)_m = ψ(A)x^(j)_m, j = 0, 1,…, m = 1,…, k. In all cases the resulting Ritz pairs are approximations to the eigenpairs of A with the special property. We provide a rather thorough convergence analysis of the approach involving all three methods as n → ∞ for the case in which A is a normal matrix. We also discuss the connections and similarities of our approach with the existing methods and approaches in the literature.     

Decomposably non-negative matrices

Let A be an mn- by - mn symmetric matrix. Partition A into m²n - by - n blocks and suppose that each of these blocks is also symmetric. Suppose that for every decomposable (rank one) tensor ν ⊗ w, we have (ν ⊗ w)^t A(ν otimes; w) ≥ 0. Here, ν is a column m-tuple and w is a column n-tuple. We study the maximum number of negative eigenvalues such a matrix can have, as well as obtaining alternative characterizations of such matrices.     

Irreducible matrices with reducible principal submatrices

Let A be a (0, 1)-matrix of order n 3 and let s_i⁰(A), i = 1, …, n, be the number of the off diagonal 0's in row and column i of A. We prove that if A is irreducible, and if all its principal submatrices of order (n − 1) are reducible, then s_i⁰(A) n − 1; i = 1, …, n. This establishes the validity of a conjecture by B. Schwarz concerning strongly connected graphs and their primal subgraphs.     

Atoms of set systems with a fixed number of pairwise unions

For a finite set system with ground set X, we let

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. An atom of H is a nonempty maximal subset C of X such that for all A H, either C A or C ∩ A = 0. We obtain a best possible upper bound for the number of atoms determined by a set system H with H = k and H H = u for all integers k and u. This answers a problem posed by Sós.     

Flat portions on the boundary of the numerical ranges of certain Toeplitz matrices

We give criterions for a flat portion to exist on the boundary of the numerical range of a matrix. A special type of Teoplitz matrices with flat portions on the boundary of its numerical range are constructed. We show that there exist 2 × 2 nilpotent matrices A₁,A₂, an n  × n nilpotent Toeplitz matrix N_n, and an n  × n cyclic permutation matrix S_n(s) such that the numbers of flat portions on the boundaries of W(A₁⊕N_n) and W(A₂⊕S_n(s)) are, respectively, 2(n - 2) and 2n.     

Flawless 0-sequences and Hilbert functions of Cohen-Macaulay integral domains

Let A = A₀A₁ be a commutative graded ring such that (i) A₀ = k a field, (ii) A = k[A₁] and (iii) dim_k A₁ < ∞. It is well known that the formal power series ∑^∞_{n = 0} (dim_kA_n)λ ⁿ is of the form (h₀ + h₁λ + + h_sλ^s)/(1 − λ)^dimA with each h_iε . We are interested in the sequence (h₀, h₁,…,h_s), called the h-vector of A, when A is a Cohen–Macaulay integral domain. In this paper, after summarizing fundamental results (Section 1), we study h-vectors of certain Gorenstein domains (Section 2) and find some examples of h-vectors arising from integrally closed level domains (Sections 3 and 4).     

Matrices with prescribed characteristic polynomial and several prescribed submatrices

Let A = [A_jk],j,k = 1,2 be a partitioned matrix. We consider the problems: find necessary and sufficient conditions for the existence of A if we prescribe (i) An and the characteristic polynomial ofA, or (ii)A₁₁,A₂₂, and the characteristic polynomial of A. We solve (i) and give a partial solution of (ii).     

A result about determinantal divisors

Let Rbe a principal ideal ringR_n the ring of n× nmatrices over R, and d_k(A) the kth determinantal divisor of Afor 1 ≤ k≤ n, where Ais any element of R_n, It is shown that if A,BεR_n, det(A) det(B:) ≠ 0, then d_k(AB) ≡ 0 mod d_k(A) d_k(B). If in addition (det(A), det(B)) = 1, then it is also shown that d_k(AB) = d_k(A) d_k(B). This provides a new proof of the multiplicativity of the Smith normal form for matrices with relatively prime determinants.     

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.     

Intersections of nest algebras in finite dimensions

If are maximal nests on a finite-dimensional Hilbert space H, the dimension of the intersection of the corresponding nest algebras is at least dim H. On the other hand, there are three maximal nests whose nest algebras intersect in the scalar operators. The dimension of the intersection of two nest algebras (corresponding to maximal nests) can be of any integer value from n to n(n+1)/2, where n=dim H. For any two maximal nests there exists a basis {f₁,f₂,…,f_n} of H and a permutation π such that and where M_i=  span{f₁,f₂,…,f_i} and N_i= span{f_π(1),f_π(2),…,f_π(i)}. The intersection of the corresponding nest algebras has minimum dimension, namely dim H, precisely when π(j)=n−j+1,1jn. Those algebras which are upper-triangular matrix incidence algebras, relative to some basis, can be characterised as intersections of certain nest algebras.     

A result on integral symmetric matrices

Let A be an integral matrix such that det A = 1 mod mA ≡ A^T mod m, where m is odd. It is shown that a symmetric integral matrix B of determinant 1 exists such that B ≡ A mod m. The result is false if m is even.