Related papers: The Dominant Eigenvalue of an Essentially Nonnegat…

The spectral theorem says that a real symmetric matrix has an orthogonal basis of eigenvectors and that, for a matrix with distinct eigenvalues, the basis is unique (up to signs). In this paper, we study the symmetric tensors with an…

Graph convolutional neural networks have been instrumental in machine learning of material properties. When representing tensorial properties, weights and descriptors of a physics-informed network must obey certain transformation rules to…

In this paper, we introduce a particular class of matrices. We study the concept of a matrix to be \emph{balanced}. We study some properties of this concept in the context of matrix operations. We examine the behaviour of various matrix…

We prove that the matrix of capacitance in electrostatics is a positive-singular matrix with a non-degenerate null eigenvalue. We explore the physical implications of this fact, and study the physical meaning of the eigenvalue problem for…

In this paper, it is proved that (strict) copositivity of a symmetric tensor $\mathcal{A}$ is equivalent to the fact that every principal sub-tensor of $\mathcal{A}$ has no a (non-positive) negative $H^{++}$-eigenvalue. The necessary and…

This work addresses the issue of large covariance matrix estimation in high-dimensional statistical analysis. Recently, improved iterative algorithms with positive-definite guarantee have been developed. However, these algorithms cannot be…

Strong ellipticity is an important property in the elasticity theory. In 2009, M-eigenvalues were introduced for the elasticity tensor. It was shown that M-eigenvalues are invariant under coordinate system choices, and the strong…

Specific definitions of the core and core-EP inverses of complex tensors are introduced. Some characterizations, representations and properties of the core and core-EP inverses are investigated. The results are verified using specific…

A Hankel tensor is called a strong Hankel tensor if the Hankel matrix generated by its generating vector is positive semi-definite. It is known that an even order strong Hankel tensor is a sum-of-squares tensor, and thus a positive…

We consider the problem of decomposing a real-valued symmetric tensor as the sum of outer products of real-valued vectors. Algebraic methods exist for computing complex-valued decompositions of symmetric tensors, but here we focus on…

Random matrices like GUE, GOE and GSE have been studied for decades and have been shown that they possess a lot of nice properties. In 2005, a new property of independent GUE random matrices is discovered by Haagerup and Thorbj{\o}rnsen in…

The spectrum of the nonbacktracking matrix associated to a network is known to contain fundamental information regarding percolation properties of the network. Indeed, the inverse of its leading eigenvalue is often used as an estimate for…

The computation of the sparse principal component of a matrix is equivalent to the identification of its principal submatrix with the largest maximum eigenvalue. Finding this optimal submatrix is what renders the problem…

We present a fully extrinsic, parametrization-free variant of tensor calculus on embedded, possibly evolving, submanifolds with boundary in arbitrary dimension and codimension. The proposed approach is component-free and, for general rank…

We study tridimensional tensors on the complex field from the point of view of hypermatrices, taking into consideration the problem of determining whether they are degenerate or not, concise or not, what is their essential format if they…

In this paper, we introduce almost (strictly) semi-positive tensors, which extend the concept of almost (strictly) semimonotone matrices. Furthermore, we provide insights into the characteristics of the entries within these almost…

We introduce $\hat{H}$-eigenvalue for $2m$-th order $n$-dimensional complex tensors. Then we determine several checkable inclusion sets for $\hat{H}$-eigenvalues and derive some criterions for the Hermitian positive definiteness…

Hu and Ye conjectured that for a $k$-th order and $n$-dimensional tensor $\mathcal{A}$ with an eigenvalue $\lambda$ and the corresponding eigenvariety $\mathcal{V}_\lambda(\mathcal{A})$, $$\mathrm{am}(\lambda) \ge \sum_{i=1}^\kappa…

As in random matrix theories, eigenvector/value distributions are important quantities of random tensors in their applications. Recently, real eigenvector/value distributions of Gaussian random tensors have been explicitly computed by…

We propose two different strategies to find eigenvalues and eigenvectors of a given, not necessarily Hermitian, matrix $A$. Our methods apply also to the case of complex eigenvalues, making the strategies interesting for applications to…