In this paper, the inverse eigenvalue problem for the bordered diagonal matrices are reconsidered whose elements are equal to zero except for the first row, the first column and the diagonal elements. The necessary and sufficient conditions for existence of a symmetric bordered diagonal matrix from special spectral data have been determined. A new algorithm to make such matrices is derived and some numerical examples are given to illustrate the efficiency of the method.

The problem of constructing a matrix by its spectral information is called inverse eigenvalue problem (IEP) which arises in a variety of applications. In this paper, we study an IEP for arrowhead matrices in different cases. The problem involves constructing of the matrix by some eigenvalues of each of the leading principal submatrices and one eigenpair. We will also investigate this problem and its variants in the cases of matrix entries being real, nonnegative, positive definite, complex and equal diagonal entries. To solve the problems, a new method to establish a relationship between the IEP and properties of symmetric and general form of matrices is developed. The necessary and sufficient conditions of the solvability of the problems  are obtained. Finally, some numerical examples are presented.

This paper presents a remarkable formula for spectral distance of a given block normal matrix GD0 = A B C D0! to set of block normal matrix GD (as same as GD0 except block D which is replaced by block D0), in which A 2 C is invertible, B 2 C n n n m with RankfGDg < n + m 1 and given eigenvalues of matrix M = D CA m n; C 2 C and D 2 C m m 1 B as z1; z2; : : :; zm where jz1j  jz2j 


 jzm1j  jzmj. Finally, an explicit formula is proven for spectral distance GD and GD0 which is expressed by the two last eigenvalues of M.

The purpose of this research is to investigate the factors affecting the formation of leading schools, so that it can be used in Iran's education. For this purpose, first, according to the research background and theoretical foundations, a conceptual framework was presented in the field of the components of an efficient and progressive school. Due to the nature of the subject, this research is an applied and developmental research and its method is of a mixed type, and includes quantitative and qualitative methods. According to the obtained information, the statistical population is about 3000 people. The sample size is estimated to be 344 people using Morgan's table. Stratified sampling method was used for sampling in this research, and 84 managers and 260 teachers participated in this study. The tool of this research is a researcher-made questionnaire. To analyze the data and answer the research questions, the correlation coefficient test and factor analysis were used. The results of the research showed that the components of management and leadership, research factors, educational factors, human resources, budget and infrastructure, technology and equipment are the main components of the formation of leading schools, and the component of educational factors has the largest contribution.

The aim of the current paper is to study a partially described inverse eigenvalue problem of  a specific symmetric  matrix, and prove some properties of such matrix. The problem includes the construction of the matrix by  the  minimal eigenvalue of all  leading principal submatrices  and eigenpair $(\lambda_2^{(n)},x)$ such that $ \lambda_2^{(n)}$ is the maximal eigenvalue of the required matrix. We investigate  conditions for the solvability of the problem, and finally an algorithm and  its numerical results are presented.

In this note we obtain a reverse version of the Haagerup Theorem. In particular, if $ A \in \mathbb{M}_{n}$ has a $ 2\times2- $ principal submatrix as $ \left[ \begin{array}{cc}1& \alpha \\\beta & 1\\\end{array}\right]$ with $ \beta \neq \bar{\alpha}, $ then $ \Vert S_{A} \Vert > 1$ where the operator $ S_{A}:\mathbb{M}_{n}\longrightarrow \mathbb{M}_{n} $ is defined by $S_{A}(B) := A \circ B $ where $ "\circ " $ stands for Schur product.