Volume 21, Issue 5
An Inverse Eigenvalue Problem for Jacobi Matrices

J. Comp. Math., 21 (2003), pp. 569-584.

Published online: 2003-10

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• Abstract

Let $T_{1,n}$ be an $n\times n$ unreduced symmetric tridiagonal matrix with eigenvalues $$\lambda_1<\lambda_2<\cdots<\lambda_n.$$ and $$W_k=\Bigg(\begin{matrix}T_{1,k-1} & 0 \\0&T_{k+1,n}\end{matrix} \Bigg)$$is an $(n-1)\times(n-1)$ submatrix by deleting the $k^{th}$ row and $k^{th}$ column, $k=1,2,\ldots,n$ from $T_n$. Let $$\mu_1\leq\mu_2\leq\cdots\leq\mu_{k-1}$$ be the eigenvalues of $T_{1,k-1}$ and $$\mu_k\leq\mu_{k+1}\leq\cdots\leq\mu_{n-1}$$ be the eigenvalues of $T_{k+1,n}$.

A new inverse eigenvalues problem has put forward as follows: How do we construct an unreduced symmetric tridiagonal matrix $T_{1,n}$, if we only know the spectral data: the eigenvalues of $T_{1,n}$, the eigenvalues of $T_{1,k-1}$ and the eigenvalues of $T_{k+1,n}$?
Namely if we only know the data: $\lambda_1,\lambda_2,\cdots,\lambda_n,\mu_1,\mu_2,\cdots,\mu_{k-1}$ and $\mu_k,\mu_{k+1},\cdots,\mu_{n-1}$ how do we find the matrix $T_{1,n}$? A necessary and sufficient condition and an algorithm of solving such problem, are given in this paper.

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@Article{JCM-21-569, author = {Jiang , Er-Xiong}, title = {An Inverse Eigenvalue Problem for Jacobi Matrices}, journal = {Journal of Computational Mathematics}, year = {2003}, volume = {21}, number = {5}, pages = {569--584}, abstract = {

Let $T_{1,n}$ be an $n\times n$ unreduced symmetric tridiagonal matrix with eigenvalues $$\lambda_1<\lambda_2<\cdots<\lambda_n.$$ and $$W_k=\Bigg(\begin{matrix}T_{1,k-1} & 0 \\0&T_{k+1,n}\end{matrix} \Bigg)$$is an $(n-1)\times(n-1)$ submatrix by deleting the $k^{th}$ row and $k^{th}$ column, $k=1,2,\ldots,n$ from $T_n$. Let $$\mu_1\leq\mu_2\leq\cdots\leq\mu_{k-1}$$ be the eigenvalues of $T_{1,k-1}$ and $$\mu_k\leq\mu_{k+1}\leq\cdots\leq\mu_{n-1}$$ be the eigenvalues of $T_{k+1,n}$.

A new inverse eigenvalues problem has put forward as follows: How do we construct an unreduced symmetric tridiagonal matrix $T_{1,n}$, if we only know the spectral data: the eigenvalues of $T_{1,n}$, the eigenvalues of $T_{1,k-1}$ and the eigenvalues of $T_{k+1,n}$?
Namely if we only know the data: $\lambda_1,\lambda_2,\cdots,\lambda_n,\mu_1,\mu_2,\cdots,\mu_{k-1}$ and $\mu_k,\mu_{k+1},\cdots,\mu_{n-1}$ how do we find the matrix $T_{1,n}$? A necessary and sufficient condition and an algorithm of solving such problem, are given in this paper.

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/8888.html} }
TY - JOUR T1 - An Inverse Eigenvalue Problem for Jacobi Matrices AU - Jiang , Er-Xiong JO - Journal of Computational Mathematics VL - 5 SP - 569 EP - 584 PY - 2003 DA - 2003/10 SN - 21 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/8888.html KW - Symmetric tridiagonal matrix, Jacobi matrix, Eigenvalue problem, Inverse eigenvalue problem. AB -

Let $T_{1,n}$ be an $n\times n$ unreduced symmetric tridiagonal matrix with eigenvalues $$\lambda_1<\lambda_2<\cdots<\lambda_n.$$ and $$W_k=\Bigg(\begin{matrix}T_{1,k-1} & 0 \\0&T_{k+1,n}\end{matrix} \Bigg)$$is an $(n-1)\times(n-1)$ submatrix by deleting the $k^{th}$ row and $k^{th}$ column, $k=1,2,\ldots,n$ from $T_n$. Let $$\mu_1\leq\mu_2\leq\cdots\leq\mu_{k-1}$$ be the eigenvalues of $T_{1,k-1}$ and $$\mu_k\leq\mu_{k+1}\leq\cdots\leq\mu_{n-1}$$ be the eigenvalues of $T_{k+1,n}$.

A new inverse eigenvalues problem has put forward as follows: How do we construct an unreduced symmetric tridiagonal matrix $T_{1,n}$, if we only know the spectral data: the eigenvalues of $T_{1,n}$, the eigenvalues of $T_{1,k-1}$ and the eigenvalues of $T_{k+1,n}$?
Namely if we only know the data: $\lambda_1,\lambda_2,\cdots,\lambda_n,\mu_1,\mu_2,\cdots,\mu_{k-1}$ and $\mu_k,\mu_{k+1},\cdots,\mu_{n-1}$ how do we find the matrix $T_{1,n}$? A necessary and sufficient condition and an algorithm of solving such problem, are given in this paper.

Er-Xiong Jiang. (1970). An Inverse Eigenvalue Problem for Jacobi Matrices. Journal of Computational Mathematics. 21 (5). 569-584. doi:
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