Linear mappings which are rank-k nonincreasing, II

We prove the following result. Let F be an infinite field of characteristic other than two. Let k be a positive integer. Let S_n(F) denote the space of all n × n symmetric matrices with entries in F, and let T:S_n(F)→S_n(F) be a linear operator. Suppose that T is rank-k nonincreasing and its image contains a matrix with rank higher than K. Then, there exist λεF and PεF^n,n such that T(A)=λPAP^t for all AεS_n(F). λ can be chosen to be 1 if F is algebraically closed and ±1 if F=R, the real field.     

Convexoid and generalized derivations

Let E,F be two Banach spaces and let S be a symmetric norm ideal of L(E,F). For AL(F) and BL(E) the generalized derivation δ_S,A,B is the operator on S that sends X to AX−XB. A bounded linear operator is said to be convexoid if its (algebraic) numerical range coincides with the convex hull of its spectrum. We show that δ_S,A,B is convexoid if and only if A and B are convexoid.     

The minimal polynomial of a matrix

Let Rbe a finite dimensional central simple algebra over a field FA be any n× n matrix over R. By using the method of matrix representation, this paper obtains the structure formula of the minimal polynomial q_A(λ) of A over F. By using q_A(λ), this paper discusses the structure of right (left) eigenvalues set of A, and obtains the necessary and sufficient condition that a matrix over a finite dimensional central division algebra is similar to a diagonal matrix.     

Large spaces of symmetric matrices of bounded rank are decomposable

Let k and n be positive integers such that k≤n. Let S_n(F) denote the space of all n×n symmetric matrices over the field F with char F≠2. A subspace L of S_n(F) is said to be a k-subspace if rank A≤k for every AεL.

Now suppose that k is even, and write k=2r. We say a k∥-subspace of S_n(F) is decomposable if there exists in Fⁿ a subspace W of dimension n-r such that x^tAx=0 for every xεWAεL.

We show here, under some mild assumptions on kn and F, that every k∥-subspace of S_n(F) of sufficiently large dimension must be decomposable. This is an analogue of a result obtained by Atkinson and Lloyd for corresponding subspaces of F^m,n.     

A characterization on n-critical economical generalized tic-tac-toe games

There is a so called generalized tic-tac-toe game playing on a finite set X with winning sets A₁, A₂,…, A_m. Two players, F and S, take in turn a previous untaken vertex of X, with F going first. The one who takes all the vertices of some winning set first wins the game. Erd s and Selfridge proved that if |A₁|=|A₂|==|A_m|=n and m<2ⁿ⁻¹, then the game is a draw. This result is best possible in the sense that once m=2ⁿ⁻¹, then there is a family A₁, A₂,…, A_m so that F can win. In this paper we characterize all those sets A₁,…, A_2n−1 so that F can win in exactly n moves. We also get similar result in the biased games.     

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.     

A matrix problem associated with computing Cohen-Macaulay type

Suppose A∈M_n×m(F), B∈M_n×t(F) for some field F. Define Г(AB) to be the set of n×n diagonal matrices D such that the column space of DA is contained in the column space of B. In this paper we determine dim Г(AB). For matrices AB of the same rank we provide an algorithm for computing dim Г(AB).     

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.     

Multiplicative properties of generalized matrix functions

We consider scalar-valued matrix functions for n×n matrices A=(a_ij) defined by Where G is a subgroup of S_n the group of permutations on n letters, and χ is a linear character of G. Two such functions are the permanent and the determinant. A function (1) is multiplicative on a semigroup S of n×n matrices if d(AB)=d(A)d(B) AB∈S.

With mild restrictions on the underlying scalar ring we show that every element of a semigroup containing the diagonal matrices on which (1) is multiplicative can have at most one nonzero diagonal(i.e., diagonal with all nonzero entries)and conversely, provided that χ is the principal character(χ≡1).     

Similarity over SL(n, F)

Let F be a finite field. It is shown that if AB axe n × n matrices with entries from F which are similar over GL(n, F), then AB are similar over SL(n, F), provided that some elementary divisor of xl- A is irreducible over F. The result remains true if F is any field such that any element of F may be represented as the norm of an element of any finite algebraic extension of F.     

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.     

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).     

The Best Lower Bound for the Permanent of a Correlation Matrix of Rank Two

If 1≤k≤n, then Cor(n,k) denotes the set of all n×n real correlation matrices of rank not exceeding k. Grone and Pierce have shown that if A∈Cor (n, n-1), then per(A)≥n/(n-1). We show that if A∈Cor(n,2), then , and that this inequality is the best possible.     

Alternative proofs of a theorem of Moyls and Marcus on the numerical range of a square matrix

Moyls and Marcus [4] showed that for n≤4,n×n an complex matrix A is normal if and only if the numerical range of A is the convex hull of the eigenvalues of A. When n≥5, there exist matrices which are not normal, but such that the numerical range is still the convex hull of the eigenvalues. Two alternative proofs of this fact are given. One proof uses the known structure of the numerical range of a 2×2 matrix. The other relies on a theorem of Motzkin and Taussky stating that a pair of Hermitian matrices with property L must commute.     

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}.     

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.     

Complementarity forms of theorems of Lyapunov and Stein, and related results

The well-known Lyapunov's theorem in matrix theory / continuous dynamical systems asserts that a (complex) square matrix A is positive stable (i.e., all eigenvalues lie in the open right-half plane) if and only if there exists a positive definite matrix X such that AX+XA^* is positive definite. In this paper, we prove a complementarity form of this theorem: A is positive stable if and only if for any Hermitian matrix Q, there exists a positive semidefinite matrix X such that AX+XA^*+Q is positive semidefinite and X[AX+XA^*+Q]=0. By considering cone complementarity problems corresponding to linear transformations of the form I−S, we show that a (complex) matrix A has all eigenvalues in the open unit disk of the complex plane if and only if for every Hermitian matrix Q, there exists a positive semidefinite matrix X such that X−AXA^*+Q is positive semidefinite and X[X−AXA^*+Q]=0. By specializing Q (to −I), we deduce the well known Stein's theorem in discrete linear dynamical systems: A has all eigenvalues in the open unit disk if and only if there exists a positive definite matrix X such that X−AXA^* is positive definite.     

Similarity to symmetric matrices over fields which are not formally real

We show that a matrix is similar to a symmetric matrix over a field of characteristic 2 if and only if the minimum polynomial of the matrix is not the product of distinct irreducible polynomials whose splitting fields are inseparable extensions. When the field is not of characteristic 2, a known theorem is generalized by considering k, the number of elementary divisors of odd degree of the n × n A: If -1 is a sum of 2^v squares and n differs from a multiple of 2^{v + 1} by at most ±k, then A is similar to a symmetric matrix.     

Extremal patterns of distinct entries in vectors in the range of a matrix

For an m × n matrix A over a field F we consider the following quantities: μ(A), the maximum multiplicity of a field element as a component of a nonzero vector in the range of A, and δ(A), the minimum number of distinct entries in a nonzero vector in the range of A. In terms of ramk(A), we describe the set of possible values of μand δ and discuss the possible relations between them. We also develop a general affine geometric structure in which the sets of values of μ and δ may be characterized linear algebraically.     

The identity as a matrix commutator

If A is an n×n matrix over a field F of positive characteristic p, then I_n=AB-BA, for some matrix B, iff p divides the size of each Jordan block of A.