Bounds on the moduli of eigenvalues of rational matrices
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866929377620525056 |
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| author | Basavaraju, Pallavi Hadimani, Shrinath Jayaraman, Sachindranath |
| author_facet | Basavaraju, Pallavi Hadimani, Shrinath Jayaraman, Sachindranath |
| contents | A rational matrix is a matrix-valued function $R(λ): \mathbb{C} \rightarrow M_p$ such that $R(λ) = \begin{bmatrix} r_{ij}(λ) \end{bmatrix}_{p\times p}$, where $r_{ij}(λ)$ are scalar complex rational functions in $λ$ for $i,j=1,2,\ldots,p$. The aim of this paper is to obtain bounds on the moduli of eigenvalues of rational matrices in terms of the moduli of their poles. To a given rational matrix $R(λ)$ we associate a block matrix $\mathcal{C}_R$ whose blocks consist of the coefficient matrices of $R(λ)$, as well as a scalar real rational function $q(x)$ whose coefficients consist of the norm of the coefficient matrices of $R(λ)$. We prove that a zero of $q(x)$ which is greater than the moduli of all the poles of $R(λ)$ will be an upper bound on the moduli of eigenvalues of $R(λ)$. Moreover, by using a block matrix associated with $q(x)$, we establish bounds on the zeros of $q(x)$, which in turn yields bounds on the moduli of eigenvalues of $R(λ)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2306_14776 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Bounds on the moduli of eigenvalues of rational matrices Basavaraju, Pallavi Hadimani, Shrinath Jayaraman, Sachindranath Spectral Theory 15A18, 15A42, 47A12, 47A56, 26C15 A rational matrix is a matrix-valued function $R(λ): \mathbb{C} \rightarrow M_p$ such that $R(λ) = \begin{bmatrix} r_{ij}(λ) \end{bmatrix}_{p\times p}$, where $r_{ij}(λ)$ are scalar complex rational functions in $λ$ for $i,j=1,2,\ldots,p$. The aim of this paper is to obtain bounds on the moduli of eigenvalues of rational matrices in terms of the moduli of their poles. To a given rational matrix $R(λ)$ we associate a block matrix $\mathcal{C}_R$ whose blocks consist of the coefficient matrices of $R(λ)$, as well as a scalar real rational function $q(x)$ whose coefficients consist of the norm of the coefficient matrices of $R(λ)$. We prove that a zero of $q(x)$ which is greater than the moduli of all the poles of $R(λ)$ will be an upper bound on the moduli of eigenvalues of $R(λ)$. Moreover, by using a block matrix associated with $q(x)$, we establish bounds on the zeros of $q(x)$, which in turn yields bounds on the moduli of eigenvalues of $R(λ)$. |
| title | Bounds on the moduli of eigenvalues of rational matrices |
| topic | Spectral Theory 15A18, 15A42, 47A12, 47A56, 26C15 |
| url | https://arxiv.org/abs/2306.14776 |