Spectral properties of sums of certain Kronecker products
The authors study spectral properties of matrices of the form \(A\otimes A + C\otimes C\) and \(I\otimes A + A\otimes I + C\otimes C\), where \(A\) and \(C\) are real square matrices, and \(I\) the identity matrix. These matrices are related to the linear endomorphisms \({\mathcal T}_{A,C}:Z\mapsto AZA^T+CZC^T\) respectively \(\widetilde{\mathcal T}_{A,C}:Z\mapsto AZ+ZA^T+CZC^T\) of the matrix algebra \(\mathbb{C}^{n\times n}\). The main results are Theorem~3 and Theorem~5. Theorem~3 asserts that the spectral radius of \({\mathcal T}_{A,C}\) equals the spectral radius of its restriction to the space \(\mathbb{S}^{n\times n}\subset\mathbb{C}^{n\times n}\) of symmetric matrices. Similarly, by Theorem~5, the spectral abscissa (\(\max\{\mathop{\mathrm{Re}}\lambda \mid \lambda\in\mathbb{C} \text{ is an eigenvalue}\}\)) of \(\widetilde{\mathcal T}_{A,C}\) is the same as the spectral abscissa of the restriction \(\widetilde{\mathcal T}_{A,C}|_{\mathbb{S}^{n\times n}}\). The presentation is rather fuzzy. Notably, Theorem~2 is really just a complicated statement (followed by an equally complicated proof) of the fact that the respective spaces of symmetric and anti-symmetric matrices are invariant under \({\mathcal T}_{A,C}\).
scientific article; zbMATH DE number 5603586
scientific article