A permuted graph matrix is a matrix V ∈ C(m+n)×m such that every row of the m×m identity matrix Im appears at least once as a row of V . Permuted graph matrices can be used in some contexts in place of orthogonal matrices, for instance when giving a basis for a subspace U ⊆ Cm+n, or to normalize matrix pencils in a suitable sense. In these applications the permuted graph matrix can be chosen wi...

This note deals with the following problem: Let A be an n X n Hermitian matrix, and Q and 0 be two n X rn (n > m > 1) matrices both with orthonormal column vectors. How do the eigenvalues of the m X m Hermitian matrix Q”AQ differ from those of the m X m Hermitian matrix QHAQ? We give a positive answer to one of the unsolved problems raised recently by Sun. In what follows, we will consider the ...

Braided m-Lie algebras induced by multiplication are introduced, which generalize Lie algebras, Lie color algebras and quantum Lie algebras. The necessary and sufficient conditions for the braided m-Lie algebras to be strict Jacobi braided Lie algebras are given. Two classes of braided m-Lie algebras are given, which are generalized matrix braided m-Lie algebras and braided m-Lie subalgebras of...

A triangular matrix algebra over a field k is defined by a triplet (R, S, M) where R and S are k-algebras and RMS is an SR-bimodule. We show that if R, S and M are finite dimensional and the global dimensions of R and S are finite, then the triangular matrix algebra corresponding to (R, S, M) is derived equivalent to the one corresponding to (S, R, DM), where DM = Homk(M, k) is the dual of M , ...

The matrix equation MA = N is considered. The lgrade of a matrix A is the the largest rank of any subdiagonal block of a symmetric partition of a square matrix. When A has lgrade d, representation results are given for MA = N with M and N having lower bandwidth d. M can be chosen to be lower triangular or unitary. A second result is that if MA = N with M and N have lower bandwidths d M and d N ...

We study the problem of reconstructing a low-rank matrix, where the input is an n × m matrix M over a field F and the goal is to reconstruct a (near-optimal) matrix M ′ that is lowrank and close to M under some distance function ∆. Furthermore, the reconstruction must be local, i.e., provides access to any desired entry of M ′ by reading only a few entries of the input M (ideally, independent o...

Nonnegative matrix factorization (NMF) is the problem of decomposing a given nonnegative n × m matrix M into a product of a nonnegative n×d matrix W and a nonnegative d×m matrix H. NMF has a wide variety of applications, including bioinformatics, chemometrics, communication complexity, machine learning, polyhedral combinatorics, among many others. A longstanding open question, posed by Cohen an...

In this supplementary material, we present proof details. 1 Notation and Conventions We use the following notation and conventions throughout the supplement. For a real n× n matrix M , we use the unadorned norm ‖M‖ to denote its spectral norm. The notation ‖M‖F refers to the Frobenius norm, ‖M‖1 is ∑ i,j |M(i, j)| and ‖M‖∞ is maxij |M(i, j)|. We will also study operators on the space of matrice...