Related papers: Eigenvalues, singular values, and the O'Shea-Sjama…
Matrix game, which is also known as two person zero sum game, is a famous model in game theory. There are some well established theories about it, such as von Neumann minimax theorem. However, almost no literature have reported the…
Diagonalization, or eigenvalue decomposition, is very useful in many areas of applied mathematics, including signal processing and quantum physics. Matrix decomposition is also a useful tool for approximating matrices as the product of a…
Recently, three numerical methods for the computation of eigenvalues of singular matrix pencils, based on a rank-completing perturbation, a rank-projection, or an augmentation were developed. We show that all three approaches can be…
We present first-order perturbation analysis of a simple eigenvalue and the corresponding right and left eigenvectors of a general square matrix, not assumed to be Hermitian or normal. The eigenvalue result is well known to a broad…
We consider the correlation functions of eigenvalues of a unidimensional chain of large random hermitian matrices. An asymptotic expression of the orthogonal polynomials allows to find new results for the correlations of eigenvalues of…
Exploiting the explicit bijection between the density of singular values and the density of eigenvalues for bi-unitarily invariant complex random matrix ensembles of finite matrix size, we aim at finding the induced probability measure on…
By using the methods of Cauchy-Binet type formula and adjugate matrix respectively, a wonderful equality relating to the elements of eigenvectors, the eigenvalues and the submatrix eigenvalues is proved in arXiv:1908.03795. In the note, we…
Eigenvectors associated with non-degenerate eigenvalues are shown to correspond to columns of the adjugate of the characteristic matrix. Degenerate eigenvalues are associated with eigenvectors that correspond to reduced complement tensors…
This paper is concerned with singular matrix difference equations of mixed order. The existence and uniqueness of initial value problems for these equations are derived, and then the classification of them is obtained with a similar…
Asymptotic behavior of the singular value decomposition (SVD) of blown up matrices and normalized blown up contingency tables exposed to Wigner-noise is investigated.It is proved that such an m\times n matrix almost surely has a constant…
We consider a product of an arbitrary number of independent rectangular Gaussian random matrices. We derive the mean densities of its eigenvalues and singular values in the thermodynamic limit, eventually verified numerically. These…
In the search for appropriate discretizations of surface theory it is crucial to preserve such fundamental properties of surfaces as their invariance with respect to transformation groups. We discuss discretizations based on M\"obius…
A new property, the strong singular value property, is introduced, developed, and utilized to study the problem of which lists of nonnegative real numbers occur as the singular values of a matrix with a prescribed zero-nonzero pattern.
The dimensions of sets of matrices of various types, with specified eigenvalue multiplicities, are determined. The dimensions of the sets of matrices with given Jordan form and with given singular value multiplicities are also found. Each…
We prove lower bounds for the smallest singular value of rectangular, multivariate Vandermonde matrices with nodes on the complex unit circle. The nodes are ``off the grid'', groups of nodes cluster, and the studied minimal singular value…
The eigenvalue distribution is investigated for matrix models related via the localization to Chern-Simons-matter theories. An integral representation of the planar resolvent is used to derive the positions of the branch points of the…
The matrix-valued spherical functions for the pair (K x K, K), K=SU(2), are studied. By restriction to the subgroup A the matrix-valued spherical functions are diagonal. For suitable set of representations we take these diagonals into a…
We consider the problem of finding nonzero eigenvalues and the corresponding eigenvectors of a matrix $AA^{\top}$, where $A$ is a special incidence matrix; This matrix can equivalently be defined based on a match relation between some…
We describe the eigenvalues and the eigenspaces of the adjacency matrices of subgraphs of the Hamming cube induced by Hamming balls, and more generally, by a union of adjacent concentric Hamming spheres. As a corollary, we extend the range…
This is a first paper by the authors dedicated to the distribution of eigenvalues for random perturbations of large bidiagonal Toeplitz matrices.