Pseudospectra, critical points and multiple eigenvalues of matrix polynomials

AbstractWe develop a general framework for perturbation analysis of matrix polynomials. More specifically, we show that the normed linear space Lm(Cn×n) of n-by-n matrix polynomials of degree at most m provides a natural framework for perturbation analysis of matrix polynomials in Lm(Cn×n). We present a family of natural norms on the space Lm(Cn×n) and show that the norms on the spaces Cm+1 and Cn×n play a crucial role in the perturbation analysis of matrix polynomials. We define pseudospectra of matrix polynomials in the general framework of the normed space Lm(Cn×n) and show that the pseudospectra of matrix polynomials well known in the literature follow as special cases. We analyze various properties of pseudospectra in the unified framework of the normed space Lm(Cn×n). We analyze critical points of backward errors of approximate eigenvalues of matrix polynomials and show that each critical point is a multiple eigenvalue of an appropriately perturbed polynomial. We show that common boundary points of components of pseudospectra of matrix polynomials are critical points. As a consequence, we show that a solution of Wilkinson’s problem for matrix polynomials can be read off from the pseudospectra of matrix polynomials

L-structure least squares solutions of reduced biquaternion matrix equations with applications

This paper presents a framework for computing the structure-constrained least squares solutions to the generalized reduced biquaternion matrix equations (RBMEs). The investigation focuses on three different matrix equations: a linear matrix equation with multiple unknown L-structures, a linear matrix equation with one unknown L-structure, and the general coupled linear matrix equations with one unknown L-structure. Our approach leverages the complex representation of reduced biquaternion matrices. To showcase the versatility of the developed framework, we utilize it to find structure-constrained solutions for complex and real matrix equations, broadening its applicability to various inverse problems. Specifically, we explore its utility in addressing partially described inverse eigenvalue problems (PDIEPs) and generalized PDIEPs. Our study concludes with numerical examples.Comment: 30 page

Special least squares solutions of the reduced biquaternion matrix equation with applications

This paper presents an efficient method for obtaining the least squares Hermitian solutions of the reduced biquaternion matrix equation $(AXB, CXD) = (E, F )$. The method leverages the real representation of reduced biquaternion matrices. Furthermore, we establish the necessary and sufficient conditions for the existence and uniqueness of the Hermitian solution, along with a general expression for it. Notably, this approach differs from the one previously developed by Yuan et al. $(2020)$, which relied on the complex representation of reduced biquaternion matrices. In contrast, our method exclusively employs real matrices and utilizes real arithmetic operations, resulting in enhanced efficiency. We also apply our developed framework to find the Hermitian solutions for the complex matrix equation $(AXB, CXD) = (E, F )$, expanding its utility in addressing inverse problems. Specifically, we investigate its effectiveness in addressing partially described inverse eigenvalue problems. Finally, we provide numerical examples to demonstrate the effectiveness of our method and its superiority over the existing approach.Comment: 25 pages, 3 figure

Respon Brokoli (Brassica oleracea) pada Berbagai Kombinasi Dosis Pupuk Organik dengan Pupuk NPK

Penelitian dalam bentuk percobaan lapangan tentang respon brokoli (Brassica oleracea) pada berbagai kombinasi dosis pupuk organik dengan pupuk NPK telah dilaksanakan di Nagari Taluak, Kabupaten Agam, Sumatera Barat sejak bulan Maret sampai Juni 2016. Tujuan dari penelitian ini adalah untuk memperoleh dosis kombinasi pupuk organik dengan pupuk NPK (15:15:15) yang tepat pada tanaman brokoli dan untuk memperoleh informasi mengenai pengaruh yang diberikan oleh kombinasi beberapa dosis pupuk organik dengan pupuk NPK (15:15:15) terhadap pertumbuhan dan hasil tanaman brokoli. Penelitian ini disusun dalam Rancangan Acak Lengkap (RAL) dengan 8 perlakuan dan 3 ulangan. Sebagai perlakuan digunakan kombinasi pupuk organik dan pupuk NPK (15:15:15) yang terdiri dari A1 = pupuk kandang ayam 10 ton/ha + 800 kg/ha NPK (15:15:15), A2 = pupuk kandang ayam 15 ton/ha + 800 kg/ha NPK (15:15:15), A3 = pupuk kandang ayam 10 ton/ha + 600 kg/ha NPK (15:15:15), A4 = Pupuk kandang ayam 15 ton/ha + 600 kg/ha NPK (15:15:15), A5 = pupuk rhizokompos 10 ton/ha + 800 kg/ha NPK (15:15:15), A6 = pupuk rhizokompos 15 ton/ha + 800 kg/ha NPK (15:15:15), A7 = pupuk rhizokompos 10 ton/ha + 600 kg/ha NPK (15:15:15), A8 = Pupuk rhizokompos 15 ton/ha + 600 kg/ha NPK (15:15:15). Berdasarkan hasil penelitian yang telah dilaksanakan dapat diambil kesimpulan bahwa penggunaan kombinasi pupuk rhizokompos 10 ton/ha + 800 kg/ha NPK (15:15:15) memberikan pengaruh terbaik terhadap pertumbuhan dan hasil tanaman brokoli

Does Entitlement Card ensure Utilization of Urban Primary Healthcare Clinic in Bangladesh?

Background: Although Bangladesh substantial­ly reduced 40% maternal death in the last deca­des, it is still challenging, especially among the ur­ban poor. The achievements are not equitable bet­ween different economic quintiles and bet­ween the rich and the urban poor. This study aims to examine the extent to which the entitle­ment card affects the utilization of maternal and child health care and identifies other factors that influence MCH services.Subjects and Method: This cross-sectional study was carried out in the working areas of the Ur­ban Primary Healthcare Project (UPHCP) in Bang­ladesh. A two-stage cluster sampling tech­ni­que was used to select the participants. A total of 3,949 women aged 15-49 years, having at least one child aged two years or less were selected for this study. The data were collected through face-to-face inter­views. The data were analyzed using multinomial logistic regression.Results: The proportion of utilization of UPH­CC was 49.9%. One-fourth (26.6%) of them fully utilized it and another 23.3% utilized it partially. Stepwise multinomial logistic regression analysis revealed that those who had an Entitlement Card from the UPHC project were 11.75 times (95% CI= 9.481, 14.549; p= 0.001) more likely to fully Utilized and 3.64 times (95% CI= 3.643, 2.911; p= 0.001) likely be utilized partially compared to non-utilizer. Respondents having no formal edu­cation utilized UPHCC fully (AOR=2.32; 95% CI= 1.46, 3.68; p= 0.001) and partially (AOR= 1.76; 95% CI= 1.12, 2.77; p= 0.014) used UPHCC. It was 3.08 (95% CI= 2.03, 4.67; p= 0.001) times for fully and 2.71 (95% CI= 1.82, 4.04; p= 0.001) times for partially utilized UPHCC compared to non-users among the primary level of education. Small family size (≤4) and monthly family in­come in the range of BDT 10,000 above were likely to utilise UPHCC. However, non-Muslims were less likely to Utilized UPHCC.Conclusion: Apart from the entitlement card, other factors such as monthly income BDT> 10,000, small family size, no formal educated mo­ther appeared to be potential predictors for utilization of the Urban Primary Health care clinic.Keywords: entitlement card, maternal care, Utilization, urban primary healthcareCorrespondence: Md Mizanur Rahman. Department of Community Medicine and Public Health, Faculty of Medicine and Health Sciences, Universiti Malaysia Sa­rawak. Email: [email protected], [email protected] of Maternal and Child Health (2020), 5(2): 213-225https://doi.org/10.26911/thejmch.2020.05.02.1