1.1 The number.dat data file from Shepard, Kilpatric, and Cunningham (1975). 11 3.1 A proximity matrix, morse digits.dat, for the 10 Morse code symbols representing the first ten digits 9.1 Order-constrained least-squares approximations to the digit proximity data of Shepard et al. (1975); the upper-triangular portion is anti-Robinson and the lower-triangular portion is strongly-anti-9.2 The 45...

This paper addresses the problems of estimating the normal covariance and precision matrices. A commutator subgroup of lower triangular matrices is considered for deriving a class of invariant estimators. The class shows inadmissibility of the best invariant and minimax estimator of the covariance matrix relative to quadratic loss. Also, in estimation of the precision matrix, a dominance result...

For a large class of ordered trees it’s possible to take results at the root and move them to an arbitrary vertex (the mutator) or to a leaf of the tree. This is the uplift principle. If you also sort by height and record the results in matrix form you end up with one or more elements of a group of infinite lower triangular matrices called the Riordan group. The main tools used are generating f...

We prove that the Ricci flow equation for left invariant metrics on Lie groups reduces to a first order ordinary differential equation for a map Q : (−a, a) → UT , where UT is the group of upper triangular matrices. We decompose the matrix Rij of Ricci tensor coordinates with respect to an orthonormal frame field Ei into a sum 1 Rij + 2 Rij + 3 Rij + 4 Rij such that, for any Ei′ = U i i′Ei with...

A fast approximate inversion method is proposed for the block lower triangular Toeplitz with tri-diagonal blocks (BL3TB) matrix. The BL3TB matrix is approximated by a block ε-circulant matrix, which can be efficiently inverted using the fast Fourier transforms. The error estimation is given to show the high accuracy of the approximation. In applications, the proposed method is employed to solve...

The Schur method for computing a matrix square root reduces the matrix to the Schur triangular form and then computes a square root of the triangular matrix. We show that by using either standard blocking or recursive blocking the computation of the square root of the triangular matrix can be made rich in matrix multiplication. Numerical experiments making appropriate use of level 3 BLAS show s...

If r = 12 and (uij) is the matrix of the U operator in the above basis, then the numbers uij satisfy a recurrence formula: there is a p × p matrix M such that uij = ∑p r,s=1Mrsui−r,j−s. Furthermore, M is skew-upper-triangular and constant on off diagonals; and the coefficients uij satisfy uij = jiuji. The case p = 2 is extensively studied in [BC05]. Here the recurrence relation is simple enough...