Novel approach to root functions of matrix polynomials with applications in differential equations and meromorphic matrix functions
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866915210334306304 |
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| author | Borogovac, Muhamed |
| author_facet | Borogovac, Muhamed |
| contents | In the first part of the paper, we address an invertible matrix polynomial $L(z)$ and its inverse $\hat{L}(z) := -L(z)^{-1}$. We present a method for obtaining a canonical set of root functions and Jordan chains of $L(z)$ through elementary transformations of the matrix $L(z)$ alone. This method provides a new and simple approach to deriving a general solution of the system of ordinary linear differential equations $L\left(\frac{d}{dt}\right)u=0$ using only elementary transformations of the corresponding matrix polynomial $L(z)$.
In the second part of the paper, given a matrix generalized Nevanlinna function $Q\in N_{κ}^{n \times n}$ and a canonical set of root functions of $\hat{Q}(z) := -Q(z)^{-1}$, we provide an algorithm to determine a specific Pontryagin space $(\mathcal{K}, [.,.])$, a specific self-adjoint operator $A:\mathcal{K}\rightarrow \mathcal{K}$ and an operator $Γ: \mathbb{C}^{n}\rightarrow \mathcal{K}$ that represent the function $Q$ in a Krein-Langer type representation. We demonstrate the main results through examples of linear systems of ODEs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_16568 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Novel approach to root functions of matrix polynomials with applications in differential equations and meromorphic matrix functions Borogovac, Muhamed Functional Analysis Complex Variables 34M03 47A56 47B50 46C20 In the first part of the paper, we address an invertible matrix polynomial $L(z)$ and its inverse $\hat{L}(z) := -L(z)^{-1}$. We present a method for obtaining a canonical set of root functions and Jordan chains of $L(z)$ through elementary transformations of the matrix $L(z)$ alone. This method provides a new and simple approach to deriving a general solution of the system of ordinary linear differential equations $L\left(\frac{d}{dt}\right)u=0$ using only elementary transformations of the corresponding matrix polynomial $L(z)$. In the second part of the paper, given a matrix generalized Nevanlinna function $Q\in N_{κ}^{n \times n}$ and a canonical set of root functions of $\hat{Q}(z) := -Q(z)^{-1}$, we provide an algorithm to determine a specific Pontryagin space $(\mathcal{K}, [.,.])$, a specific self-adjoint operator $A:\mathcal{K}\rightarrow \mathcal{K}$ and an operator $Γ: \mathbb{C}^{n}\rightarrow \mathcal{K}$ that represent the function $Q$ in a Krein-Langer type representation. We demonstrate the main results through examples of linear systems of ODEs. |
| title | Novel approach to root functions of matrix polynomials with applications in differential equations and meromorphic matrix functions |
| topic | Functional Analysis Complex Variables 34M03 47A56 47B50 46C20 |
| url | https://arxiv.org/abs/2407.16568 |