Construction of symmetric pentadiagonal matrix from three interlacing spectrum

Document Type : Research Paper

Authors

1 Department of Mathematics, Sahand University of Technology, Tabriz, IRAN

2 Department of Mathematics, Sahand University of Technology, Tabriz, Iran

Abstract

‎In this paper‎, ‎we introduce a new algorithm for constructing a‎ ‎symmetric pentadiagonal matrix by using three interlacing spectrum‎, ‎say (λi)i=1n‎, ‎(μi)i=1n and (νi)i=1n‎ ‎such that‎
‎‎‎0<λ1<μ1<λ2<μ2<...<λn<μn,‎‎μ1<ν1<μ2<ν2<...<μn<νn‎,‎‎
‎where (λi)i=1n are the eigenvalues of pentadiagonal‎ ‎matrix A‎, ‎(μi)i=1n are the eigenvalues of A∗ (the‎   ‎matrix A∗ differs from A only in the (1,1) entry) and‎ ‎(νi)i=1n are the eigenvalues of A∗∗ (the matrix‎ ‎A∗∗ differs from A∗ only in the (2,2) entry)‎. ‎From the‎
‎interlacing spectrum‎, ‎we find the first and second columns of‎ ‎eigenvectors‎. ‎Sufficient conditions for the solvability of the problem‎ ‎are given‎. ‎Then we construct the pentadiagonal matrix A from these‎ ‎eigenvectors and given eigenvalues by using the block Lanczos algorithm‎. ‎We‎ ‎also give an example to demonstrate the efficiency of the algorithm‎.

Keywords

References

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Journal of Algebra and Related Topics
Volume 10, Issue 2
December 2022
Pages 89-98

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