An Efficient Construction of Secure Network Co ding

Under the assumption that the wiretapper can get at most r(r n) independent messages, Cai et al. showed that any rate n multicast code can be modified to another secure network code with transmitting rate n- r by a properly chosen matrix Q~(-1). They also gave the construction for searching such an n × n nonsingular matrix Q. In this paper, we find that their method implies an efficient construction of Q. That is to say, Q can be taken as a special block lower triangular matrix with diagonal subblocks being the(n- r) ×(n- r)and r × r identity matrices, respectively. Moreover, complexity analysis is made to show the efficiency of the specific construction.     

Semiscalar equivalence and the factorization of polynomial matrices

One considers the problem of the factorization of polynomial matrices over an arbitrary field in connection with their reducibility by semiscalar equivalent transformations to triangular form with the invariant factors along the principal diagonal. In particular, one establishes a criterion for the representability of a polynomial matrix in the form of a product of factors (the first of which is unital), the product of the canonical diagonal forms of which is equal to the canonical diagonal form of the given matrix. There is given also a method for the construction of such factorizations.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 5, pp. 644–649, May, 1990.     

On the matrix equation $$ X\bar X = \bar XX $$

It is shown that every solution to the equation $ X\bar X = \bar XX $ X\bar X = \bar XX can be reduced by a real orthogonal similarity transformation to a block triangular form with diagonal blocks of orders one and two. If the solution X is a normal matrix, then its block triangular form is actually a block diagonal. In this case, the form of the diagonal blocks is found, yielding new proof of the recent results of Goodson and Horn.     

On the triangular factorization of positive operators

An operator integral, referred to as the amplitude integral (AI) and used in the BC-method (based on boundarycontrol theory) for solving inverse problems, is systematically studied. For a continuous operator and two families of increasing subspaces, the continual analog of the matrix diagonal in the form of an AI is introduced. The convergence of the AI is discussed. An example of an operator with no diagonal is provided. The role of the diagonal in the problem of triangular factorization is elucidated. The well-known result of matrix theory stating the uniqueness of triangular factorization with a prescribed diagonal is extended. It is shown that the corresponding factor can be represented in the AI form. The correspondence between the AI and the classical representation of the triangular factor of an operator that is a sum of the identity and a compact operator is established.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 239, 1997, pp. 45–60.This work was supported by INTAS under grant INTAS 93-1815 and by the Russian Foundation for Basic Research under grant 96-01-00666. The authors are grateful to Prof. S. N. Naboko for fruitful discussions and consultations.     

Domination in Generalized Cayley Graph of Commutative Rings

Let $R$ be a commutative ring with identity and $n$ be a natural number. The generalized Cayley graph of $R$, denoted by $Γ^n_R$, is the graph whose vertex set is $R^n$\{0} and two distinct vertices $X$ and $Y$ are adjacent if and only if there exists an $n×n$ lower triangular matrix $A$ over $R$ whose entries on the main diagonal are non-zero such that $AX^T=Y^T$ or $AY^T=X^T$, where for a matrix $B$, $B^T$ is the matrix transpose of $B$. In this paper, we give some basic properties of$Γ^n_R$ and we determine the domination parameters of$Γ^n_R$.     

Similarity of block diagonal and block triangular matrices

It is shown that if a block triangular matrix is similar to its block diagonal part, then the similarity matrix can be chosen of the block triangular form. An analogous statement is proved for equivalent matrices. For the simplest case of 2×2 block matrices these results were obtained by W.Roth [1]. It is shown that all these results do not admit a generalization for the infinite dimensional case.     

There exist square real matrices of an arbitrary order n ≥ 2880 which are not DOTU-matrices

One finds conditions under which a given real square matrix A cannot be represented in the form A=D0TU, where D, 0, T are diagonal, orthogonal, and triangular real matrices, respectively, while U is a unimodular integral matrix. From these one derives the statement formulated in the title.     

Matricial formulae for partitions

The exponential of the triangular matrix whose entries in the diagonal at distance n from the principal diagonal are all equal to the sum of the inverses of the divisors of n is the triangular matrix whose entries in the diagonal at distance n from the principal diagonal are all equal to the number of partitions of n. A similar result holds for all pairs of sequences satisfying a special mutual recurrence.     

On Permutations to Block Triangular Form

This note clarifies some points relating to the permutationof a matrix to block triangular form. In particular, the blocktriangular form of any matrix which may be permuted to havea zero-free diagonal is shown to be unique apart from certainwell-defined trivial permutations.     

有强法式的体上矩阵

设D为除环,A∈Dn×n,则可用初等变换将λI-A化简为对角阵A= diag(1,…,1,φ1,…,φr),其中(?)i为D上首1多项式并且φ1|…|φr.如果这个对角阵A在形状上是唯一的,则称A是有强法式的矩阵.本文应用中心原子因子与初等因子给出了体上有强法式的矩阵的本质刻画,给出了体上矩阵有强法式的一些充要条件.     

Normal dilatation of triangular matrices

LetR be a (real or complex) triangular matrix of ordern, say, an upper triangular matrix. Is it true that there exists a normaln×n matrixA whose upper triangle coincides with the upper triangle ofR? The answer to this question is “yes” and is obvious in the following cases: (1)R is real; (2)R is a complex matrix with a real or a pure imaginary main diagonal, and moreover, all the diagonal entries ofR belong to a straight line. The answer is also in the affirmative (although it is not so obvious) for any matrixR of order 2. However, even forn=3 this problem remains unsolved. In this paper it is shown that the answer is in the affirmative also for 3×3 matrices.     

Cones, graphs and optimal scalings of matrices

Characterizations are given of the optimal scalings of a complex square matrix within its diagonal similarity class and its restricted diagonal equivalence class with respect to the maximum element norm. The characterizations are in terms of a finite number of products, principally circuit and diagonal products. The proofs proceed by reducing the optimal scaling problems from the multiplicative matrix level in succession to an additive matrix level, a graph theoretic level, and a geometric level involving duality theorems for cones. At the geometric level, the diagonal similarity and the restricted diagonal equivalence problems are unified.     

Analysis of incomplete factorizations for a nine-point approximation to a convection–diffusion model problem

We study the stability of zero-fill incomplete LU factorizations of a nine-point coefficient matrix arising from a high-order compact discretisation of a two-dimensional constant-coefficient convection–diffusion problem. Nonlinear recurrences for computing entries of the lower and upper triangular matrices are derived and we show that the sequence of diagonal entries of the lower triangular factor is unconditionally convergent. A theoretical estimate of the limiting value is derived and we show that this estimate is a good predictor of the computed value. The unconditional convergence of the diagonal sequence of the lower triangular factor to a positive limit implies that the incomplete factorization process never encounters a zero pivot and that the other diagonal sequences are also convergent. The characteristic polynomials associated with the lower and upper triangular solves that occur during the preconditioning step are studied and conditions for the stability of the triangular solves are derived in terms of the entries of the tridiagonal matrices appearing in the lower and upper subdiagonals of the block triangular system matrix and a triplet of parameters which completely determines the solution of the nonlinear recursions. Results of ILU-preconditioned GMRES iterations and the effects of orderings on their convergence are also described.     

Similarity transformations of decomposable matrix polynomials and related questions

A relationship is found between the similarity transformations of decomposable matrix polynomials with relatively prime elementary divisors and the equivalence transformations of the corresponding matrices with scalar entries. Matrices with scalar entries are classified with respect to equivalence transformations based on direct sums of lower triangular almost Toeplitz matrices. This solves the similarity problem for a special class of finite matrix sets over the field of complex numbers. Eventually, this problem reduces to the one of special diagonal equivalence between matrices. Invariants of this equivalence are found.     

Representation of unimodular matrices in the form DOTU for n=3

A proof is given of the representation of any unimodular matrix of order three as the product of a diagonal, orthogonal, triangular and unimodular integral one.Translated from Matematicheskie Zametki, Vol. 18, No. 2, pp. 213–221, August, 1975.The author expresses gratitude for critical remarks to B. N. Delone and S. S. Ryshkov.     

Block Triangularization of Algebras of Matrices

This paper is concerned with the interdependence of the irreducible constituents of an algebra of n × n matrices over a field F. It is shown that there is a similarity transformation reducing the algebra to a block triangular form in which, ateach pair of diagonal places, the blocks either are always equal or may be occupied by any entries from the corresponding irreducible constituents. A recent theorem of Kaplan-sky is extended as an application of this result.     

Ergodic reduction of random products of two-by-two matrices

We consider a random product of two-by-two matrices of determinant one over an abstract dynamical system. When the two Lyapunov exponents are distinct, Oseledets’ theorem asserts that the matrix cocycle is cohomologous to a diagonal matrix cocycle. When they are equal, we show that the cocycle is conjugate to one of three cases: a rotation matrix cocycle, an upper triangular matrix cocycle, or a diagonal matrix cocycle modulo a rotation by π/2.     

Reduction of matrix polynomials to a special quasitriangular form

We find necessary and sufficient conditions for polynomials with matrix coefficients over an arbitrary field to be reducible via a similarity transformation to block triangular form with regular diagonal blocks of maximum degree. Translated fromMatematichni Metody i Fiziko-Mekhanichni Polya, Vol. 38, 1995.     

四元数矩阵方程AXAH=B的最小二乘解

引入了四元数矩阵范数的概念，通过使用四无数矩阵的奇异值分解，给出了四元数矩阵方程AXA^H＝B在最小二乘意义下的Hermitian解以及Skew-Hermitian解．     

Estimation of the Separation of Two Matrices (Ⅱ)

In this paper we give a lower bound of the separation $sep_F(A,B)$ of two diagonalizable matrices A and B. The key to finding the lower bound of $sep_F(A,B)$ is to find an upper bound for the condition number of a transformation matrix Q which transforms a diagonalizable matrix A to a diagonal form. The obtained lower bound of $sep_F(A,B)$ involves the eigenvalues of A and B as well as the departures form the normality $\delta_F(A)$ and $\delta_F(B)$.