This note describes a technique for generating large non-singular matrices with blocks of full rank. Our motivation to construct such matrices arises in the white-box implementation of cryptographic algorithms with S-boxes.

In this paper we characterize those linear operators on general matrices that preserve singular values and displacement rank. We also characterize those linear operators on Hermitian matrices that preserve eigenvalues and displacement inertia.

then σ is a singular value of A and u and v are corresponding left and right singular vectors, respectively. (For generality it is assumed that the matrices here are complex, although given these results, the analogs for real matrices are obvious.) If, for a given positive singular value, there are exactly t linearly independent corresponding right singular vectors and t linearly independent co...

Solving large scale system of Simultaneous Linear Equations (SLE) has been (and continue to be) a major challenging problem for many real-world engineering and science applications. Solving SLE with singular coefficient matrices arises from various engineering and sciences applications [1]-[6]. In this paper, efficient numerical procedures for finding the generalized (or pseudo) inverse of a ge...

This paper presents an O(n2 log n) algorithm for computing the symmetric singular value decomposition of square Hankel matrices of order n, in contrast with existing O(n3) SVD algorithms. The algorithm consists of two stages: first, a complex square Hankel matrix is reduced to a complex symmetric tridiagonal matrix using the block Lanczos method in O(n2 log n) flops; second, the singular values...

Ensembles of isotropic random matrices are defined by the invariance of the probability measure under the left (and right) multiplication by an arbitrary unitary matrix. We show that the multiplication of large isotropic random matrices is spectrally commutative and self-averaging in the limit of infinite matrix size N→∞. The notion of spectral commutativity means that the eigenvalue density of...

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×n matrix almost surely has a constant number of large singular values (of order √ mn), while the rest of the singular values are of order √ m + n as m,n → ∞. Concentration results of Alon at al. for the...

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.

In this paper we propose a fast structure-preserving algorithm for computing the singular value decomposition of quaternion matrices. The algorithm is based on the structurepreserving bidiagonalization of the real counterpart for quaternion matrices by applying orthogonal JRS-symplectic matrices. The algorithm is efficient and numerically stable. 2014 Elsevier Inc. All rights reserved.