For the low rank approximation of time-dependent data matrices and of solutions to matrix differential equations, an increment-based computational approach is proposed and analyzed. In this method, the derivative is projected onto the tangent space of the manifold of rank-r matrices at the current approximation. With an appropriate decomposition of rank-r matrices and their tangent matrices, th...

We address some theoretical guarantees for Schatten-p quasi-norm minimization (p ∈ (0, 1]) in recovering low-rank matrices from compressed linear measurements. Firstly, using null space properties of the measuring operator, we provide a sufficient condition for exact recovery of low-rank matrices. This condition guarantees unique recovery of matrices of ranks equal or larger than what is guaran...

In this paper, we study subspace embedding problem and obtain the following results: 1. We extend the results of approximate matrix multiplication from the Frobenius norm to the spectral norm. Assume matrices A and B both have at most r stable rank and r̃ rank, respectively. Let S be a subspace embedding matrix with l rows which depends on stable rank, then with high probability, we have ‖ASSB−A...

The article is devoted to different aspects of the question: ”What can be done with a matrix by a low rank perturbation?” It is proved that one can change a geometrically simple spectrum drastically by a rank 1 perturbation, but the situation is quite different if one restricts oneself to normal matrices. Also the Jordan normal form of a perturbed matrix is discussed. It is proved that with res...

For generic lower triangular matrices, A, we prove that A ij = P d q=1 H iq G jq for i > j is equivalent to A = M ?1 N where M and N are d+1 banded matrices. A lower triangular matrix A is input balanced of order/rank d if there exists a rank-d matrix B such that AA = I ? BB. We prove that if A is triangular input balanced then generically, A = M ?1 N where M and N are d + 1 banded matrices. Th...

With Qq,n the distribution of n minus the rank of a matrix Mn chosen uniformly from Mat(n, q), the collection of all n × n matrices over the finite field Fq of size q ≥ 2, and Qq the distributional limit of Qq,n as n→∞, we apply Stein’s method to prove the total variation bound 1 8qn+1 ≤ ||Qq,n −Qq||TV ≤ 3 qn+1 . In addition, we obtain similar sharp results for the rank distributions of symmetr...

Abstract. Let G be an undirected graph on n vertices and let S(G) be the set of all real symmetric n×n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. Let mr(G) denote the minimum rank of all matrices in S(G), and mr+(G) the minimum rank of all positive semidefinite matrices in S(G). All graphs G with mr(G) = 2 and mr+(G) = k are chara...

We show that the maximal cp-rank of n×n completely positive matrices is attained at a positive-definite matrix on the boundary of the cone of n×n completely positive matrices, thus answering a long standing question. We also show that the maximal cp-rank of 5×5 matrices equals six, which proves the famous Drew-JohnsonLoewy conjecture (1994) for matrices of this order. In addition we present a s...

3 Connection matrices 9 3.1 The connection matrix of a graph parameter . . . . . . . . . . . . . . . . . . . . 9 3.2 The rank of connection matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.3 Connection matrices of homomorphisms . . . . . . . . . . . . . . . . . . . . . . . 12 3.4 The exact rank of connection matrices for homomorphisms . . . . . . . . . . . . 13 3.5 Extensions...