Showing papers on "Triangular matrix published in 1985"

A commutant lifting theorem for triangular matrices with diverse applications

Abstract: A lifting theorem for representations of the algebra of block upper triangular matrices is proved analogous to the commutatant lifting theorem of Sarason and Sz.-Nagy and Foias. Included is a description by linear fractional maps of all solutions of the lifting problem. This first part of the paper contains the statements of the main results and applications to matrix interpolation and completion problems.

Strong laws of large numbers for products of random matrices

Abstract: This work, on products of random matrices, is inspired by papers of Furstenberg and Kesten (Ann. Math. Statist. 31 (1960), 457-469) and Furstenberg (Trans. Amer. Math. Soc. 108 (1963), 377-428). In particular, a formula was known for almost sure limits for normalized products of random matrices in terms of a stationary measure. However, no explicit computational techniques were known for these limits, and little was known about the stationary measures. We prove two main theorems. The first assumes that the random matrices are upper triangular and computes the almost sure limits in question. For the second, we assume the random matrices are 2 x 2 and Bernoulli, i.e., random matrices whose support is two points. Then the second theorem gives an asymptotic result for the almost sure limits, with rates of convergence in some cases.

Auxiliary Storage Methods for Solving Finite Element Systems

Abstract: In this paper we consider the direct solution of the system of linear equations $Ax = b$, where A is a large, sparse, symmetric and positive definite matrix. This system is solved using the Cholesky method by factoring A into $R^T R$, where R is an upper triangular matrix, and then solving $R^T y = b$ and $Rx = y$. We are particularly interested in the case where A is so large that auxiliary storage must be used to store the Cholesky factor R.The approach we use, which fully exploits the sparsity of A, is based on partitioning the given system into subsystems, each of which is sufficiently small that it can be processed in a relatively small amount of main memory. We give a detailed analysis of the input/output (I/O) traffic generated when our method is applied to a model n by n grid problem. The grid is partitioned using an incomplete nested dissection, which is a minor modification of the standard nested dissection ordering. Our analysis shows that if we have $O(n^2 )$ main memory, then the total I/O traffic is $O(n^2 \log n)$. This result implies that the $O(n^3 )$ numerical computations dominate the I/O traffic. A widely used method for solving such large linear systems is the band method, in which zeros outside the band of A are exploited. The I/O traffic and arithmetic operations in this case are given by $O(n^3 )$ and $O(n^4 )$, respectively.We also consider an improvement for our basic strategy which reduces the I/O traffic to the extent that it is dominated by writing the Cholesky factor alone. This enhancement reduces the I/O traffic associated with storing intermediate results from $O(n^2 \log n)$ to $O(n^2 \log \log n)$. Numerical experiments are also provided to illustrate the performance of our algorithms.

On simultaneous reduction of families of matrices to triangular or diagonal form by unitary congruences

Abstract: Let {Ai }and {Bi } be two given families of n-by-n matrices. We give conditions under which there is a unitary U such that every matrix UAiU 1 is upper triangular. We give conditions, weaker than the classical conditions of commutativity of the whole family, under which there is a unitary U such that every matrix UAjU ∗ is upper triangular. We also give conditions under which there is one single unitary U such that every UAiU 1 and every UBjU ∗ is upper triangular. We give necessary and sufficient conditions for simultaneous unitary reduction to diagonal form in this way when all the Aj's are complex symmetric and all theBj 's are Hermitian.

T-ideals containing all matrix polynomial identities

Abstract: Any T-ideal, having as a proper subset all polynomial identities of the 2 × 2 matrix algebra, contains a product of commutators of length 2. The main purpose of the paper is to estimate the number of factors of this product. Equivalently, the class of nilpotence of the Jacobson radical of the corresponding relatively free algebra is bounded. Similarresults are established for Lie and Jordan 2 × 2 matrix algebras. As a corollary,the Specht property is obtained for the T-ideal of the Jordan algebra of all symmetric matrices of second order.

Homology of integral upper-triangular matrices

Abstract: We calculate the homology of the multiplicative group of integral upper-triangular n x n matrices at all primes p > n 1. 1. Group homology. For any ring A, let Gln(A) be the general linear group of invertible n x n matrices over A, and Un(A) the subgroup of upper-triangular matrices with ones on the diagonal. The purpose of this note is to prove Theorem 1.1, which gives a calculation of the Eilenberg-Mac Lane group homology of Un(Z) at all primes p > n 1. (This group homology is the same as the homology of the compact nilmanifold Un(R)/Un(Z).) For any permutation a in the symmetric group Sn, let Ma E Gl n(A) denote the matrix with the property that Ma bi = ba(i), where b, is the column vector with one in the ith coordinate and zero in the others. Let Un-(A) be the group of lower-triangular matrices in Gl n(A) with ones on the diagonal, and Un? the intersection (M, 1Un-(A)M,) n Un(A). It is easy to see that Unb(A) is the subgroup of Un(A) given by matrices which have zero in row i, columnj whenever i u(j)} We will denote this cardinality by 1(a). Note that the permutation a is uniquely determined by the set L( a). Let Z(p) stand for the localization of the ring Z at the prime p. 1.1 THEOREM. If p > n 1, then H*(Un(Z),Z(p)) is the free Z(p)-module on the classes Ca, a E Sn. The proof of Theorem 1.1 is in two steps. Let Un denote Un(Z). 1.2 PROPOSITION. The cycles Ca, a E S, generate a free Z-summand of U*(Un, Z) of rank n!. Received by the editors April 17, 1984. 1980 Mathematics Subject Classification. Primary 55R35; Secondary 55S30, 17B56, 20H25. ?1985 American Mathematical Society 0002-9939/85 $1.00 + $.25 per page

Convergence of Chahine's nonlinear relaxation inversion method used for limb viewing remote sensing

Abstract: The application of Chahine's (1970) inversion technique to remote sensing problems utilizing the limb viewing geometry is discussed. The problem considered here involves occultation-type measurements and limb radiance-type measurements from either spacecraft or balloon platforms. The kernel matrix of the inversion problem is either an upper or lower triangular matrix. It is demonstrated that the Chahine inversion technique always converges, provided the diagonal elements of the kernel matrix are nonzero.

Upper triangular invariants

Abstract: Abstract We modify the construction of the mod 2 Dyer-Lashof (co)-algebra to obtain a (co)-algebra W which is (also) unstable over the Steenrod algebra A*. W has canonical sub-coalgebras W[k] such that the hom-dual W[k:]* is isomorphic as an A-algebra to the ring of upper triangular invariants in ℤ/2ℤ [x1, . . . , xk].

Systolic matrix multiplier for digital data processing

Abstract: The multiplier, useable especially for matrixing and dematrixing of digital video signals, makes it possible to carry out a numerical operation of the type A*X + B*Y + C*Z. It comprises a first assembly MAT1 consisting of a systolic calculating matrix with n x n elementary cells (n being the number of coding bits) each carrying out an elementary matrix operation A*X + B*Y + C*Z, feeding a second assembly intended for propagating carries and results of elementary calculations for obtaining high-significance bits, the second assembly MAT2 in turn feeding a third assembly consisting of a triangular matrix with n register lines for presentation of high-significant bits on corresponding outputs.

Extension of rsa cryptosystems to matrix rings

Abstract: A generalization of the RSA cryptosystem in the ring of matrices over Z/mZ is presented. It is shown that factorization of the modulus m is needed to compute the exponent of the group formed by either non-singular matrix messages or upper triangular matrices including diagonal elements thus offering the same level of security as the RSA system. The latter method employing the triangular matrices as messages seems to be more practical than the use of arbitrary non-singular matrix messages. The scheme is as suitable for privacy and authentication as its predecessor.

Nilpotent approximations of control systems

Abstract: We show how to constructively approximate an affine control system x = X0(x) + u1 (t)X1(x) + ... + uk(t)Xk(x) on IRn by a control system of this same form, and also on IRn, for which the describing vector fields generate a nilpotent Lie algebra, or at least an important subalgebra of the algebra they generate, is nilpotent.

Rapid lu decomposer

Abstract: PURPOSE:To obtain a parallel arithmetic unit executing the Lu decomposition of simultaneous linear equations using a general non-dense matrix as coefficient by driving the whole cells synchronously. CONSTITUTION:A search cell array 8 arranging cells having comparing and latching functions two-dimensionally is combined with a product summing cell array 9 arranging cells having a function summing products one-dimensionally. How to couple the memories is different in accordance with the calculation of an upper triangular part and that of a lower triangular part. In case of the calculation of the upper triangular part, a column No. memory 4 for the upper triangular part and its element value memory 5 are connected to a search cell 8 and a product summing cell 9 respectively. The non-''0'' element position and an element value of the upper triangular part of the matrix A are stored in the memories 4, 5 respectively at the start of calculation, but at the end of the calculation, the non-''0'' element position and non-''0'' element value of a matrix U are stored in the memories 4, 5 respectively. Therefore, element to be a ''0'' element in the matrix A, but to be a non-''0'' element in the matrix U (called the generation of fill in), should be precedently checked and an address for that element should be secured.

The QRPS algorithm: A generalization of the QR algorithm for the singular values decomposition of rectangular matrices

Abstract: A generalization of the QR algorithm proposed by Francis [2] for square matrices is introduced for the singular values decomposition of arbitrary rectangular matrices. Geometrically the algorithm means the subsequent orthogonalization of the image of orthonormal bases produced in the course of the iteration. Our purpose is to show how to get a series of lower triangular matrices by alternate orthogonal-upper triangular decompositions in different dimensions and to prove the convergence of this series.

An efficient algorithm for antenna synthesis updating following null-constraint changes

Abstract: The procedure to maximise the array signal to noise ratio with null constraints involves an optimisation problem that can be solved efficiently using a modified Cholesky decomposition (UD) technique. Following changes in the main lobe and/or null positions, the optimal element weight vector can be updated without the need for a new complete matrix inversion. Some properties of the UD technique can be utilised such that the updating algorithm reprocesses only a part of the unit triangular matrix U. Proper ordering of matrix entries can minimise the dimension of the updated part.

On the strong matrix summability of derived fourier series

Abstract: Strong summability with respect to a triangular matrix has been defined and applied to derived Fourier series yielding a result which extends some known results under a general criterion