On Monic Inverse Quadratic Eigenvalue Problems

Inverse quadratic eigenvalue problem refers to solving the matrices M,C and K that satisfies certain property and structure requirements under the given information of eigen-values and eigenvectors,so that the characteristic information given is the eigeninformation of quadratic matrices polynomial Q（λ）=λ²I+λC+K.Inverse quadratic eigenvalue prob-lem often requires that the matrices M,C and K are symmetric,positive definite and tridi-agonal,which is widely used in physics and has high practical value.The research on this problem has high theoretical and practical significance.The paper mainly studies the monic inverse quadratic eigenvalue problem,that is,the matric M is required to be unitary matric I.The paper mainly studies the following two problems:Problem 1:Given k eigenvaluesλ₁,λ₂,...,λ_kand corresponding eigenvectors x₁,x₂,...,x_k,find real symmetric matrices C and K,make eigenvaluesλ₁,λ₂,...,λ_kand corresponding eigen-vectors x₁,x₂,...,x_kare eigenvalues of quadratic matrices polynomial Q（λ）=λ²I+λC+K.Problem 2:Given k eigenvaluesλ₁,λ₂,...,λ_kand corresponding eigenvectors x₁,x₂,...,x_k,find real pentagonal symmetric matrices C and K,make eigenvaluesλ₁,λ₂,...,λ_kand corre-sponding eigenvectors x₁,x₂,...,x_kare eigenvalues of quadratic matrices polynomial Q（λ）=λ²I+λC+K.For problem 1,this paper gives the condition of the problem is solvable and the expres-sion of solution in the case k≤n,and studies on the case of K positive semidefinite.this paper dtudies the condition of the problem is solvable in the case k>n,and give the expres-sion of solution.For problem 2,this paper studies the condition of the problem has pentagonal solu-tion,and give the expression of solution.this paper also give an algorithm to solve this prob-lem with K positive definite.For problem 1 and problem 2,this paper gives corresponding algorithms and numerci-cal examples which show that algorithms is effective and feasible.