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 $(\lambda_i)_{i=1}^n$‎, ‎$(\mu_i)_{i=1}^n$ and $(\nu_i)_{i=1}^n$‎ ‎such that‎
‎\begin{eqnarray*}‎
‎0<\lambda_1<\mu_1<\lambda_2<\mu_2<...<\lambda_n<\mu_n,\\‎
‎\mu_1<\nu_1<\mu_2<\nu_2<...<\mu_n<\nu_n‎,
‎\end{eqnarray*}‎
‎where $(\lambda_i)_{i=1}^n$ are the eigenvalues of pentadiagonal‎ ‎matrix $A$‎, ‎$(\mu_i)_{i=1}^n$ are the eigenvalues of $A^*$ (the‎   ‎matrix $A^*$ differs from $A$ only in the $(1,1)$ entry) and‎ ‎$(\nu_i)_{i=1}^n$ 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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