Dynamics of two by two symmetric matrices of trace zero
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909139374964736 |
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| author | Mukherjee, Arijit |
| author_facet | Mukherjee, Arijit |
| contents | In this paper, we describe the entire structure of the vector space $Sym_2^0$ of all symmetric matrices of size $2$ having trace zero. This is motivated by the geometrical interpretation of any arbitrary element of $Sym_2^0$. We further study the orbits and stable sets of these elements. As an application of the obtained structure of $Sym_2^0$, we obtain the symmetric matrices of size $2$, trace of whose product with any trace zero symmetric matrix is zero. Finally some well known trigonometric formulas are interpreted geometrically incorporating the anatomy of $Sym_2^0$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_11201 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Dynamics of two by two symmetric matrices of trace zero Mukherjee, Arijit Rings and Algebras 15A63, 15B10, 97G60, 97H60 In this paper, we describe the entire structure of the vector space $Sym_2^0$ of all symmetric matrices of size $2$ having trace zero. This is motivated by the geometrical interpretation of any arbitrary element of $Sym_2^0$. We further study the orbits and stable sets of these elements. As an application of the obtained structure of $Sym_2^0$, we obtain the symmetric matrices of size $2$, trace of whose product with any trace zero symmetric matrix is zero. Finally some well known trigonometric formulas are interpreted geometrically incorporating the anatomy of $Sym_2^0$. |
| title | Dynamics of two by two symmetric matrices of trace zero |
| topic | Rings and Algebras 15A63, 15B10, 97G60, 97H60 |
| url | https://arxiv.org/abs/2403.11201 |