Inverse Eigenvalue Problems for Proportional Matrices

The inverse eigenvalue problems for a kind of proportional matrix are considered. How to construct a proportional matrix A from the minimal and maximal eigenvalues λ₁⁽ⁿ⁾ < … < λ₁⁽²⁾ < λ₁⁽¹⁾ < λ₂⁽²⁾ < … < λ_n⁽ⁿ⁾ (2n-1(n ≥ 2)real numbers)of its leading principal submatrices. The necessary and sufficient condition for the unique solutions and the sufficient condition for the solvability of the problems are derived. In addition the expressions are given. Finally, two algorithmic procedures are generated to verify the correctness of conclusions.