In [Rank-Width and Well-Quasi-Ordering of Skew-Symmetric or Symmetric Matrices, arXiv:1007.3807v1] Oum proved that, for a fixed finite field F, any infinite sequenceM1,M2, . . . of (skew) symmetric matrices over F of bounded F-rank-width has a pair i < j, such that Mi is isomorphic to a principal submatrix of a principal pivot transform of Mj . We generalise this result to σ-symmetric matrices ...

Matrix factorizations are a popular tool to mine regularities from data. There are many ways to interpret the factorizations, but one particularly suited for data mining utilizes the fact that a matrix product can be interpreted as a sum of rank-1 matrices. Then the factorization of a matrix becomes the task of finding a small number of rank-1 matrices, sum of which is a good representation of ...

The (0, 1)-matrix A of order n is a tournament matrix provided A + A + I = J, where I is the identity matrix, and J = Jn is the all 1’s matrix of order n. It was shown by de Caen and Michael that the rank of a tournament matrix A of order n over a field of characteristic p satisfies rankp(A) ≥ (n − 1)/2 with equality if and only if n is odd and AA = O. This article shows that the rank of a tour...

In this paper, complex singular Wishart matrices and their applications are investigated. In particular, a volume element on the space of positive semidefinite m×m complex matrices of rank n < m is introduced and some transformation properties are established. The Jacobian for the change of variables in the singular value decomposition of general m × n complex matrices is derived. Then the dens...

We propose a novel convex relaxation of sparse principal subspace estimation based on the convex hull of rank-d projection matrices (the Fantope). The convex problem can be solved efficiently using alternating direction method of multipliers (ADMM). We establish a near-optimal convergence rate, in terms of the sparsity, ambient dimension, and sample size, for estimation of the principal subspac...

The minimum (symmetric) rank of a simple graph G over a field F is the smallest possible rank among all symmetric matrices over F whose ijth entry (for i 6= j) is nonzero whenever {i, j} is an edge in G and is zero otherwise. The problem of determining minimum (symmetric) rank has been studied extensively. We define the minimum skew rank of a simple graph G to be the smallest possible rank amon...

We consider the rank of a class sparse Boolean matrices size $n \times n$. In particular, we show that probability such matrix has full rank, and is thus invertible, positive constant with value about 0.2574 for large $n$. The arise as vertex-edge incidence 1-out 3-uniform hypergraphs. result null space bounded in expectation can be contrasted results usual models matrices, based on random $k$-...

1 Basics 2 Linear Equations 3 Linear Maps 4 Rank One Matrices 5 Algebra of Matrices 6 Eigenvalues and Eigenvectors 7 Inner Products and Quadratic Forms 8 Norms and Metrics 9 Projections and Reflections 10 Similar Matrices 11 Symmetric and Self-adjoint Maps 12 Orthogonal and Unitary Maps 13 Normal Matrices 14 Symplectic Maps 15 Differential Equations 16 Least Squares 17 Markov Chains 18 The Expo...

In this note, we use a natural desingularization of the conormal variety of the variety of (n × n)-symmetric matrices of rank at most r to find a general formula for the algebraic degree in semidefinite programming. 1. The algebraic degree in semidefinite programming Let P be a general projective space of symmetric (n×n)−matrices up to scalar multiples, and let Yr ⊂ P m be the subvariety of mat...