A space of matrix of low rank is a vector space of rectangular matrices whose maximum rank is stricly smaller than the number of rows and the numbers of columns. Among these are the compression spaces, where the rank condition is garanteed by a rectangular hole of 0’s of appropriate size. Spaces of matrices are naturally encoded by linear matrices. The latter have a double existence: over the r...

In this paper an implicit (double) shifted QR-method for computing the eigenvalues of companion and fellow matrices will be presented. Companion and fellow matrices are Hessenberg matrices, that can be decomposed into the sum of a unitary and a rank 1 matrix. The Hessenberg, the unitary as well as the rank 1 structures are preserved under a step of the QR-method. This makes these matrices suita...

We derive an explicit count for the number of singular n × n Hankel (Toeplitz) matrices whose entries range over a finite field with q elements by observing the execution of the Berlekamp/ Massey algorithm on its elements. Our method yields explicit counts also when some entries above or on the anti-diagonal (diagonal) are fixed. For example, the number of singular n × n Toeplitz matrices with ...

We propose a constructive algorithm, called the tensor-based Kronecker product (KP) singular value decomposition (TKPSVD), that decomposes an arbitrary real matrix A into a finite sum of KP terms with an arbitrary number of d factors, namely A = ∑R j=1 σj A dj ⊗ · · · ⊗A1j . The algorithm relies on reshaping and permuting the original matrix into a d-way tensor, after which its tensor-train ran...

Structured low-rank approximation is the problem of minimizing a weighted Frobenius distance to a given matrix among all matrices of fixed rank in a linear space of matrices. We study the critical points of this optimization problem using algebraic geometry. A particular focus lies on Hankel matrices, Sylvester matrices and generic linear spaces.

The first 2 PARI-GP files below give the sets of vertices and edges of the Voronoi graph first in dimensions 2 to 6, then in dimension 7. The numerical data are extracted from Jaquet’s thesis [Ja]. The third file, based on Chapters 9 and 14 of [Mar], is devoted to minimal classes in dimensions 2 to 4. We present below a short account of Voronoi’s theory and minimal classes. 1. The perfection ra...

For any n×n matrix A , we use the joint higher rank numerical range, Λk(A, . . . ,Am) , to define the higher rank numerical hull of A . We characterize the higher rank numerical hulls of Hermitian matrices. Also, the higher rank numerical hulls of unitary matrices are studied. Mathematics subject classification (2010): 15A60,81P68.

We show that determining Kapranov rank of tropical matrices is not only NP-hard over any field but if Diophantine equations over the rational numbers is undecidable, determining Kapranov rank over the rational numbers is also undecidable. We prove that Kapranov rank of tropical matrices is not bounded in terms of tropical rank, answering a question of Develin, Santos, and Sturmfels [4].

The SPQR RANK package contains routines that calculate the numerical rank of large, sparse, numerically rank-deficient matrices. The routines can also calculate orthonormal bases for numerical null spaces, approximate pseudoinverse solutions to least squares problems involving rankdeficient matrices, and basic solutions to these problems. The algorithms are based on SPQR from SuiteSparseQR (ACM...

Let Mn(R) and Sn(R) be the spaces of n × n real matrices and real symmetric matrices respectively. We continue to study d(n, n − 2,R): the minimal number such that every -dimensional subspace of Sn(R) contains a nonzero matrix of rank n−2 or less. We show that d(4, 2,R) = 5 and obtain some upper bounds and monotonicity properties of d(n, n − 2,R). We give upper bounds for the dimensions of n − ...