Joint spectral radius and forbidden products
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910500794662912 |
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| author | Vladimirov, Alexander |
| author_facet | Vladimirov, Alexander |
| contents | We address the problem of finite products that attain the joint spectral radius of a finite number of square matrices. Up to date the problem of existence of "forbidden products" remained open. We prove that the product $AABABABB$ (together with its circular shifts and their mirror images) never delivers the strict maximum to the joint spectral radius if we restrict consideration to pairs $\{A,B\}$ of real $2\by 2$ matrices. Under this restriction circular shifts and their mirror images constitute the class of isospectral products and hence they all have the same spectral radius for any pair $\{A,B\}$ of $2\by 2$ matrices, even complex. For pairs of complex matrices we have numerical evidence that $AABABABB$ is still a fobidden product. A couple of binary words that encode products from this isospectral class also happen to be the shortest forbidden patterns in the parametric family of double rotations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_17524 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Joint spectral radius and forbidden products Vladimirov, Alexander Optimization and Control We address the problem of finite products that attain the joint spectral radius of a finite number of square matrices. Up to date the problem of existence of "forbidden products" remained open. We prove that the product $AABABABB$ (together with its circular shifts and their mirror images) never delivers the strict maximum to the joint spectral radius if we restrict consideration to pairs $\{A,B\}$ of real $2\by 2$ matrices. Under this restriction circular shifts and their mirror images constitute the class of isospectral products and hence they all have the same spectral radius for any pair $\{A,B\}$ of $2\by 2$ matrices, even complex. For pairs of complex matrices we have numerical evidence that $AABABABB$ is still a fobidden product. A couple of binary words that encode products from this isospectral class also happen to be the shortest forbidden patterns in the parametric family of double rotations. |
| title | Joint spectral radius and forbidden products |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2406.17524 |