Abstract: In this paper, a system of linear matrix equations
is considered. A new necessary and sufficient condition for the
consistency of the equations is derived by means of the generalized
singular-value decomposition, and the explicit representation of the
general solution is provided.
Abstract: There is a world-wide need for the development of sustainable management strategies to control pest infestation and the development of phosphine (PH3) resistance in lesser grain borer (Rhyzopertha dominica). Computer simulation models can provide a relatively fast, safe and inexpensive way to weigh the merits of various management options. However, the usefulness of simulation models relies on the accurate estimation of important model parameters, such as mortality. Concentration and time of exposure are both important in determining mortality in response to a toxic agent. Recent research indicated the existence of two resistance phenotypes in R. dominica in Australia, weak and strong, and revealed that the presence of resistance alleles at two loci confers strong resistance, thus motivating the construction of a two-locus model of resistance. Experimental data sets on purified pest strains, each corresponding to a single genotype of our two-locus model, were also available. Hence it became possible to explicitly include mortalities of the different genotypes in the model. In this paper we described how we used two generalized linear models (GLM), probit and logistic models, to fit the available experimental data sets. We used a direct algebraic approach generalized inverse matrix technique, rather than the traditional maximum likelihood estimation, to estimate the model parameters. The results show that both probit and logistic models fit the data sets well but the former is much better in terms of small least squares (numerical) errors. Meanwhile, the generalized inverse matrix technique achieved similar accuracy results to those from the maximum likelihood estimation, but is less time consuming and computationally demanding.
Abstract: Let T and S be a subspace of Cn and Cm, respectively.
Then for A ∈ Cm×n satisfied AT ⊕ S = Cm, the generalized
inverse A(2)
T,S is given by A(2)
T,S = (PS⊥APT )†. In this paper, a
finite formulae is presented to compute generalized inverse A(2)
T,S
under the concept of restricted inner product, which defined as <
A,B >T,S=< PS⊥APT,B > for the A,B ∈ Cm×n. By this
iterative method, when taken the initial matrix X0 = PTA∗PS⊥, the
generalized inverse A(2)
T,S can be obtained within at most mn iteration
steps in absence of roundoff errors. Finally given numerical example
is shown that the iterative formulae is quite efficient.
Abstract: In this paper, we first give the representation of the general solution of the following inverse eigenvalue problem (IEP): Given X ∈ Rn×p and a diagonal matrix Λ ∈ Rp×p, find nontrivial real-valued symmetric arrow-head matrices A and B such that AXΛ = BX. We then consider an optimal approximation problem: Given real-valued symmetric arrow-head matrices A, ˜ B˜ ∈ Rn×n, find (A, ˆ Bˆ) ∈ SE such that Aˆ − A˜2 + Bˆ − B˜2 = min(A,B)∈SE (A−A˜2 +B −B˜2), where SE is the solution set of IEP. We show that the optimal approximation solution (A, ˆ Bˆ) is unique and derive an explicit formula for it.