A class of inverse eigenvalue problems for real symmetric banded matrices with odd bandwidth

作者：Li, Jicheng; Dong, Liqiang^*; Li, Guo

来源：Linear Algebra and Its Applications, 2018, 541: 131-162.

DOI：10.1016/j.laa.2017.12.004

• Eigenvalue
• Inverse eigenvalue problems
• Real symmetric banded matrices
• Leading principal submatrix
• Block Lanczos algorithms

In this paper, we propose and discuss a class of inverse eigen-value problems for real symmetric banded matrices with odd bandwidth. For an odd 2p + 1 with a positive integer p, the problem is to construct an n x n real symmetric banded matrix with bandwidth 2p + 1 whose m x in leading principal sub matrix is a given m x m real symmetric banded matrix with bandwidth 2p + 1 and spectrum is a given set of real numbers {lambda(i)}(i=1)(n), where the number of distinct real numbers of {lambda(i)}(i=1)(n)is 2k when m = pk, or 2k+1 when pk < m < p(k+1), where m, n and k are positive integers and m < n. We point out that the well-known double dimensional (DD) problem is a special case of our proposed inverse eigenvalue problems. The necessary and sufficient condition for the solvability of the above inverse eigenvalue problem is derived, and the target real symmetric banded matrix can be constructed by the block Lanczos algorithms when the inverse eigenvalue problem is solvable. Several numerical examples show that our algorithms are feasible. Some concluding remarks are introduced.

• 出版日期2018-3-15
• 单位西安交通大学; 深圳大学