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.

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.