Spectral bounds for certain special type of rational matrices
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866917002083303424 |
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| author | Basavaraju, Pallavi Hadimani, Shrinath Jayaraman, Sachindranath |
| author_facet | Basavaraju, Pallavi Hadimani, Shrinath Jayaraman, Sachindranath |
| contents | The aim of this manuscript is to derive bounds on the moduli of eigenvalues of special type of rational matrices of the form $T(λ) = \displaystyle -B_0 +Iλ+\frac{B_1}{λ-α_1}+ \dots+ \frac{B_m}{λ-α_m}$, where $B_i$'s are $n \times n$ complex matrices and $α_i$'s are distinct complex numbers, using the following methods: $(1)$ an upper bound is obtained using the Bauer-Fike theorem for complex matrices on an associated block matrix $C_T$ of the given rational matrix $T(λ)$, $(2)$ a lower bound is obtained in terms of a zero of a scalar real rational function $p(x)$ associated with $T(λ)$, using Rouch$\text{é}$'s theorem for matrix-valued functions and $(3)$ an upper bound is also obtained using a numerical radius inequality for a block matrix $C_q$ associated with another scalar real rational function $q(x)$ corresponding to $T(λ)$. These bounds are compared when the coefficients are unitary matrices. Numerical examples are given to illustrate the results obtained. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2302_02894 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Spectral bounds for certain special type of rational matrices Basavaraju, Pallavi Hadimani, Shrinath Jayaraman, Sachindranath Spectral Theory Numerical Analysis 15A18, 15A42, 15A22, 93B18, 47A56 The aim of this manuscript is to derive bounds on the moduli of eigenvalues of special type of rational matrices of the form $T(λ) = \displaystyle -B_0 +Iλ+\frac{B_1}{λ-α_1}+ \dots+ \frac{B_m}{λ-α_m}$, where $B_i$'s are $n \times n$ complex matrices and $α_i$'s are distinct complex numbers, using the following methods: $(1)$ an upper bound is obtained using the Bauer-Fike theorem for complex matrices on an associated block matrix $C_T$ of the given rational matrix $T(λ)$, $(2)$ a lower bound is obtained in terms of a zero of a scalar real rational function $p(x)$ associated with $T(λ)$, using Rouch$\text{é}$'s theorem for matrix-valued functions and $(3)$ an upper bound is also obtained using a numerical radius inequality for a block matrix $C_q$ associated with another scalar real rational function $q(x)$ corresponding to $T(λ)$. These bounds are compared when the coefficients are unitary matrices. Numerical examples are given to illustrate the results obtained. |
| title | Spectral bounds for certain special type of rational matrices |
| topic | Spectral Theory Numerical Analysis 15A18, 15A42, 15A22, 93B18, 47A56 |
| url | https://arxiv.org/abs/2302.02894 |