Linear Algebra and its Applications最新文献

In this paper, we obtain a combinatorial expression for the Perron root and eigenvectors of a non-negative integer matrix using techniques from symbolic dynamics. We associate such a matrix with a multigraph and consider the edge shift corresponding to it. This gives rise to a collection of forbidden words $F$ which correspond to the non-existence of an edge between two vertices, and a collection of repeated words $R$ with multiplicities which correspond to multiple edges between two vertices. In general, for given collections $F$ of forbidden words and $R$ of repeated words with pre-assigned multiplicities, we construct a generalized language as a multiset. A combinatorial expression that enumerates the number of words of fixed length in this generalized language gives the Perron root and eigenvectors of the adjacency matrix. We also obtain conditions under which such a generalized language is a language of an edge shift.

In this note, we give simple proofs of two recent extensions of the Hua determinant inequality. We also answer a question of Poon affirmatively on a Young type inequality, which appears in his investigation of inequalities on eigenvalues arising from the Hua determinant inequality.

Closed formulas for the multilinear rank of trifocal Grassmann tensors are obtained. An alternative process to the standard HOSVD is introduced for the computation of the core of trifocal Grassmann tensors. Both of these results are obtained, under natural genericity conditions, leveraging the canonical form for these tensors, obtained by the same authors in a previous work. A gallery of explicit examples is also included.

In the current work, we study the eigenvalue distribution results of a class of non-normal matrix-sequences which may be viewed as a low rank perturbation, depending on a parameter $\beta >1$, of the basic Toeplitz matrix-sequence ${\left\{T_n\right(e^{i\theta }\left)\right\}}_{n\in N}$, $i^2=-1$. The latter of which has obviously all eigenvalues equal to zero for any matrix order n, while for the matrix-sequence under consideration we will show a strong clustering on the complex unit circle. A detailed discussion on the outliers is also provided. The problem appears mathematically innocent, but it is indeed quite challenging since all the classical machinery for deducing the eigenvalue clustering does not cover the considered case. In the derivations, we resort to a trick used for the spectral analysis of the Google matrix plus several tools from complex analysis. We only mention that the problem is not an academic curiosity and in fact stems from problems in dynamical systems and number theory. Additionally, we also provide numerical experiments in high precision, a distribution analysis in the Weyl sense concerning both eigenvalues and singular values is given, and more results are sketched for the limit case of $\beta =1$.

In 2010, Nikiforov conjectured that for $\ell \ge 2$ and n sufficiently large, $S_{n,\ell -1}^1$ is the unique graph with the maximum spectral radius over all n-vertex $C_{2\ell }$-free graphs. In 2022, Cioabǎ, Desai and Tait solved this conjecture. The theta graph ${\Theta }_{t,\ell }$ consists of two vertices joined by t vertex-disjoint paths, each of length ℓ. Particularly, ${\Theta }_{2,\ell }\cong C_{2\ell }$. In this paper, we characterize the unique extremal graph which attains the maximum spectral radius among all ${\Theta }_{t,\ell }$-free graphs of order n, where $t,\ell \ge 3$ and n is sufficiently large.