We describe recent work of Klyachko, Totaro, Knutson, and Tao, that characterizes eigenvalues of sums of Hermitian matrices, and decomposition of tensor products of representations of GLn(C). We explain related applications to invariant factors of products of matrices, intersections in Grassmann varieties, and singular values of sums and products of arbitrary matrices.

Some properties that nominally involve the eigenvalues of Gaussian Unitary Ensemble (GUE) can instead be phrased in terms of singular values. By discarding the signs of the eigenvalues, we gain access to a surprising decomposition: the singular values of the GUE are distributed as the union of the singular values of two independent ensembles of Laguerre type. This independence is remarkable giv...

In the present paper, we introduce some singular integral operators, singular quadrature operators and discretization matrices of singular integral equations with Hilbert kernel. These results both improve the classical theory of singular integral equations and develop the theory of singular quadrature with Hilbert kernel. Then by using them a unified framework for various collocation methods o...

Cointegration analysis involves the solution of a generalized eigenproblem involving moment matrices and inverted moment matrices. These formulae are unsuitable for actual computations because the condition numbers of the resulting matrices are unnecessarily increased. Our note discusses how to use the structure of the problem to achieve numerically stable computations, based on QR and singular...

In this paper we study conditions to guarantee the nonnegativity of a discretetime singular control system. A first approach can be found in the literature for general systems by using the entire coefficient matrices. Also, the particular case of matrices of index 1 has been treated by using a block decomposition and the group-projector of the matrix that gives the singularity to the system. In...

We describe recent work of Klyachko, Totaro, Knutson, and Tao, that characterizes eigenvalues of sums of Hermitian matrices, and decomposition of tensor products of representations of GLn(C). We explain related applications to invariant factors of products of matrices, intersections in Grassmann varieties, and singular values of sums and products of arbitrary matrices.

Singular value decomposition (SVD) is a widely used technique for dimensionality reduction and computation of basis vectors. In many applications, especially in fluid mechanics, the matrices are dense, but low-rank matrices. In these cases, a truncated SVD corresponding to the most significant singular values is sufficient. In this paper, we propose a tree based merge-and-truncate algorithm to ...

In this paper, aiming at solving the bidiagonal SVD problem, a classical divide-andconquer (DC) algorithm is modified, which needs to compute the SVD of broken arrow matrices by solving secular equations. The main cost of DC lies in the updating of singular vectors, which involves two matrix-matrix multiplications. We find that the singular vector matrices of a broken arrow matrix are Cauchy-li...

We compare the block-Lanczos and the Davidson methods for computing a basis of a singular subspace associated with the smallest singular values of large matrices. We introduce a simple modiication on the preconditioning step of Davidson's method which appears to be eecient on a range of large sparse matrices. Calcul du sous espace singulier associ e aux plus petites valeurs singuli eres de matr...

The Gaussian wiretap channel model is studied when there are multiple antennas at the sender, the receiver and the eavesdropper and the channel matrices are fixed and known to all the terminals. The secrecy capacity is characterized using a Sato type upper bound. A computable characterization of the capacity is provided. The secrecy capacity is studied in several interesting regimes. The high S...