On matrices stochastically similar to matrices with equal diagonal elements

A necessary and sufficient condition for a matrix to be stochastically similar to a matrix with equal diagonal elements is obtained Aand B are called Stochastically similar if B=SAS^{- 1} where S is quasi-stochastic i.e., all row sums of .S are I. An inverse elementary divisor problem for quasi-stochastic matrices is also considered.     

Some involutory similarities

Matrices A,B over an arbitrary field F, when given to be similar to each other, are shown to be involutorily similar (over F) to each other (i.e.B = CAC^-1for some C = C^-1over F) in the following cases: (1)B= aI - Afor some a ε F and (2) B = A^-1. Result (2) for the cases where char F ≠ 2 is essentially a 1966 result of Wonenburger.     

The construction of irreducible normalized representations of the general linear group

A method is described for constructing in an explicit form an irreducible representation T of M_n(F), the set of all n × n matrices over the real or complex field F, satisfying the condition T(A^*)=T^*(A) for all A∈M_n(F).     

Linear transformations on symmetric matrices II

Let Tbe a linear mapping on the space of n× nsymmetric matrices over a field Fof characteristic not equal to two. We obtain the structure of Tfor the following cases:(i) Tpreserves matrices of rank less than three; (ii) Tpreserves nonzero matrices of rank less than K + 1 where Kis a fixed positive integer less than nand Fis algebraically closed; (iii) Tpreserves rank Kmatrices where Kis a fixed odd integer and Fis algebraically closed.     

Essentially doubly stochastic matrices

In this paper we extend the general theory of essentially doubly stochastic (e.d.s.) matrices begun in earlier papers in this series. We complete the investigation in one direction by characterizing all of the algebra isomorphisms between the algebra of e.d.s. matrices of order n over a field F,E_n(F), and the total algebra of matrices of order n - 1over F,M_n-1(F) We then develop some of the theory when Fis a field with an involution. We show that for any e,f§Fof norm 1,e≠f every e.d.s. matrix in E_n(F) is a unique e.d.s. sum of an e.d.s. e-hermitian matrix and an e.d.s. f-hermitian matrix in E_n(F) Next, we completely determine the cases for which there exists an above-mentioned matrix algebra isomorphism preserving adjoints. Finally, we consider cogredience in E_n(F) and show that when such an adjoint-preserving isomorphism exists and char M_n(F) two e.d.s. e-hermitian matrices which are cogredient in M_n(F) are also cogredient in E_n(F). Using this result, we obtain simple canonical forms for cogredience of e.d.s. e-hermitian matrices in E_n(F) when Fsatisfies special conditions. This ncludes the e.d.s. skew-symmetric matrices, where the involution is trivial and E = -1.     

The maximun of i(XAX-1+B)

Let F be an algebraically closed field. We denote by i(A) the number of invariant polynomials of a square matrix A, which are different from 1. For A,B any n×n matrices over F, we calculate the maximum of i(XAX^-1+B), where X runs over the set of all non-singular n×n matrices over F.     

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.     

A note on compact graphs

An undirected simple graph G is called compact iff its adjacency matrix A is such that the polytope S(A) of doubly stochastic matrices X which commute with A has integral-valued extremal points only. We show that the isomorphism problem for compact graphs is polynomial. Furthermore, we prove that if a graph G is compact, then a certain naive polynomial heuristic applied to G and any partner G′ decides correctly whether G and G′ are isomorphic or not. In the last section we discuss some compactness preserving operations on graphs.     

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.     

Nonchordal positive semidefinite stochastic matrices

Let K_nbe the convex set of n×npositive semidefinite doubly stochastic matrices. If Aε k_n, the graph of A,G(A), is the graph on n vertices with (i,j) an edge if a_ij ≠ 0i≠ j. We are concerned with the extreme points in K_n. In many cases, the rank of Aand G(A) are enough to determine whether A is extreme in K_n. This is true, in particular, if G(A)is a special kind of nonchordal graph, i.e., if no two cycles in G(A)have a common edge.     

Some problems and results concerning the spectrums of stochastic matrices

Let Abe an n× nnonnegative matrix. Under suitable conditions there exist diagonal matrices Dand Ewith positive main diagonals such that DAEis doubly stochastic. In this paper we discuss some problems concerning the spectrums of Aand DAEwhen Ais row stochastic.     

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.     

A characterization and representation of the generalized inverse AT,S and its applications

This paper presents an explicit expression for the generalized inverse A⁽²⁾_T,S. Based on this, we established the characterization, the representation theorem and the limiting process for A⁽²⁾_T,S. As an application, we estimate the error bound of the iterative method for approximating A⁽²⁾_T,S.     

Doubly stochastic matrices whose powers eventually stop

In this note we characterize doubly stochastic matrices A whose powers A,A²,A³,… eventually stop, i.e., A^p=A^p+1= for some positive integer p. The characterization enables us to determine the set of all such matrices.     

Hypernormal matrices

An nxn matrix A is hypernormal if APA^*=A^*PA for all permutation matrices P. We shall explain how to construct hypernormal matrices.     

Products of involutory matrices. I

It is shown that, for every integer ≥1 and every field F, each n×n matrix over F of determinant ±1 is the product of four involutory matrices over F. Products of three ×n involutory matrices over F are characterized for the special cases where n≤4 or F has prime order ≤5. It is also shown for every field F that every matrix over F of determinant ±1 having no more than two nontrivial invariant factors is a product of three involutory matrices over F.     

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.     

Two linear transformations each tridiagonal with respect to an eigenbasis of the other

Let denote a field, and let V denote a vector space over with finite positive dimension. We consider a pair of linear transformations A:V→V and A^*:V→V satisfying both conditions below:

1. [(i)] There exists a basis for V with respect to which the matrix representing A is diagonal and the matrix representing A^* is irreducible tridiagonal.

2. [(ii)] There exists a basis for V with respect to which the matrix representing A^* is diagonal and the matrix representing A is irreducible tridiagonal.

We call such a pair a Leonard pair on V. Refining this notion a bit, we introduce the concept of a Leonard system. We give a complete classification of Leonard systems. Integral to our proof is the following result. We show that for any Leonard pair A,A^* on V, there exists a sequence of scalars β,γ,γ^*,,^* taken from such that both

where [r,s] means rs−sr. The sequence is uniquely determined by the Leonard pair if the dimension of V is at least 4. We conclude by showing how Leonard systems correspond to q-Racah and related polynomials from the Askey scheme.     

Numerical method for solving 3D inverse scattering problems

Let q(x) L²(D), D R³ is a bounded domain, q = 0 outside D, q is real-valued. Assume that A(\Gj;\t';,\Gj;,k) A(\Gj;\t';,\Gj), the scattering amplitude, is known for all \Gj;|t',\Gj; S², S² is the unit sphere, an d a fixed k \r>0. These data determine q(x) uniquely and a numerical method is given for computing q(x).     

Some recent results on drazin monotonicty of property n martices

If a square matrix has a nonnegarive power it is called a property-n matrix. In a recent paper [12] Werner derived some interesting necessary and sufficient conditions for a property n property-n matrix to be Drazin-montone. In particular, it was shown that a property -n matrix with ind(A) = k is Drazin-monotone if and only if A^2k+1 is weak-r-monotone. Our principal aim here is to show how this result can he strengthened considerably. To tackle this problem we also present several further results on the structure of Drazin-monotone (property-n) matrices.