一类特殊矩阵特征值反问题

The Inverse Eigenvalue Problems for a Class of Special Matrix

引入带比例Jacobi矩阵特征值反问题，讨论在给定的带比例Jacobi矩阵中嵌入一行一列使之成为新的带比例Jacobi矩阵，并具有指定的最小和最大特征值．通过特征多项式的递推关系，证明了此问题有唯一解的充要条件，并利用这些结果解决了具有2n-1个极端特征值的Jacobi和带比例Jacobi矩阵的逆特征值问题．讨论了利用带比例Jacobi矩阵Jn的顺序主子阵的所有特征值构造Jn，最后提供数值算法和算例验证了定理的正确性．

The inverse eigenvalue problems for proportional Jacobi matrices are introduced,and construcing a new proportional Jacobi matrix with specified minimal and maximal eigenvalues by embedding a row and column in a given proportional Jacobi matrix. With the recurrence relations of characteristic polynomials, the necessary and sufficient conditions of existing the unique solutions for this problem are proved,furthermore based on these results,the inverse eigenvalue problems of the Jacobi and proportional Jacobi matrices with 2n1extreme eigen- vlaues are solved. The construction of proportional Jacobi matrix Jn with eigenvalues of its all leading principle submatrices are also discussed, finally the numerical algorithms and examples are given to verify the correctness of the theories.