Set inertia-preserving matrices*†

Let V be the set of n×n complex or real general matrices, Hermitian matrices, symmetric matrices, positive definite (resp. semi-definite) matrices, diagonal matrices, or upper triangular matrices. Fix k∈Z\\01. We characterize linear maps ψ:V→V that satisfy ψAk=ψAk on an open neighborhood S of In in V. The k-power preservers are necessarily k-potent preservers, and k=2 corresponds to Jordan homomorphisms. Applying the results, we characterize maps ϕ,ψ:V→V that satisfy “trϕAψBk=trABk for all A∈V, B∈S, and ψ is linear” or “trϕAψBk=trABk for all A,B∈S and both ϕ and ψ are linear.” The characterizations systematically extend existing results in literature, and they have many applications in areas like quantum information theory. Some structural theorems and power series over matrices are widely used in our characterizations.

Componentwise analysis of direct factorization of real symmetric and hermitian matrices

Impact of aging on nanoparticle toxicity in real matrices is scarcely investigated due to a lack of suitable methodologies. Herein, the toxicity of pristine and aged silver nanoparticles (Ag NPs) to a bioluminescent Pseudomonas putida bioreporter was measured in spiked crude and final wastewater samples (CWs and FWs, respectively) collected from four wastewater treatment plants (WWTPs). Results showed lower toxicity of pristine Ag NPs in CWs than in FWs. The effect of the matrix on the eventual Ag NP toxicity was related to multiple physico-chemical parameters (biological oxygen demand (BOD), chemical oxygen demand (COD), total suspended solids (TSS) pH, ammonia, sulfide and chloride) based on a multivariate analysis. However, no collection site effect was concluded. Aged Ag NPs (up to eight weeks) were found less toxic than pristine Ag NPs in CWs; evident increased aggregation and decreased dissolution were associated with aging. However, Ag NPs exhibited consistent toxicity in FWs despite aging; comparable results were obtained in artificial wastewater (AW) simulating effluent. The study demonstrates the potency of performing nanoparticle acute toxicity testing in real and complex matrices such as wastewaters using relevant bacterial bioreporters.

The bipolar decomposition

Thompson's famous theorems on singular values–diagonal elements of the orbit of an n×n matrix A under the action (1) U(n) ⊗ U(n) where A is complex, (2) SO(n) ⊗ SO(n), where A is real, (3) O(n) ⊗ O(n) where A is real are fully examined. Coupled with Kostant's result, the real semi-simple Lie algebra son, n yields (2) and hence (3) and the sufficient part (the hard part) of (1). In other words, the curious subtracted term(s) are well explained. Although the diagonal elements corresponding to (1) do not form a convex set in Cn, the projection of the diagonal elements into Rn (or iRn) is convex and the characterization of the projection is related to weak majorization. An elementary proof is given for this hidden convexity result. Equivalent statements in terms of the Hadamard product are also given. The real simple Lie algebra sun, n shows that such a convexity result fits into the framework of Kostant's result. Convexity properties and torus relations are studied. Thompson's results on the convex hull of matrices (complex or real) with prescribed singular values, as well as Hermitian matrices (real symmetric matrices) with prescribed eigenvalues, are generalized in the context of Lie theory. Also considered are the real simple Lie algebras sop, q and sop, q, p < q, which yield the rectangular cases. It is proved that the real part and the imaginary part of the diagonal elements of complex symmetric matrices with prescribed singular values are identical to a convex set in Rn and the characterization is related to weak majorization. The convex hull of complex symmetric matrices and the convex hull of complex skew symmetric matrices with prescribed singular values are given. Some questions are asked.

2-Adic valuations of certain ratios of products of factorials and applications

Maximum permanent and Hermitian matrices

Finding a positive definite linear combination of two Hermitian matrices

Let V be a space of matrices which is one of the following three types. (I) V = Vm,n , the space of all m × n matrices with entries in a field F. We assume that F is an infinite field, and that m≤n. (II) V=, the space of all n × n complex hermitian matrices. (III) , the space of all n × n real symmetric matrices. Given a positive integer k and a linear map T: V → V, we say T is rank-k nonincreasing provided that p(T(A)) ≤ k whenever p(A) = k, where p(A) denotes the rank of A. Our purpose is to obtain some results concerning rank-k nonincreasing maps.

In this section, we seek to tie up some loose ends. First, although we have stated all the definitions and proved all the results in §§8–9 as if they applied only to real (n × n) matrices A, the truth is that just about everything said in §§8–9 about real matrices has its analogue in the theory of complex matrices. Let us go through §§8–9 point by point and isolate the (few) instances where realness of A makes a difference.

The numerical range of a matrix, according to Kippenhahn, is determined by a hyperbolic determinantal form of linear Hermitian matrices associated to the matrix. On the other hand, using Riemann theta functions, Helton and Vinnikov confirmed that a hyperbolic form always admits a determinantal representation of linear real symmetric matrices. The Riemann matrix of the hyperbolic curve plays the main role in the existence of real symmetric matrices. In this article, we implement computations of the Riemann matrix and the Abel–Jacobi variety of the hyperbolic curve associated to a determinantal polynomial of a matrix. Further, we prove that the lattice of the Abel–Jacobi variety is decomposed into the direct sum of two orthogonal lattices for some 4×4 Toeplitz matrices.

DDVV-type inequality for Hermitian matrices

The projective power method for the algebraic eigenvalue problem

Shotgun metagenomics sequencing is a powerful tool for the characterization of complex biological matrices, enabling analysis of prokaryotic and eukaryotic organisms and viruses in a single experiment, with the possibility of reconstructing de novo the whole metagenome or a set of genes of interest. One of the main factors limiting the use of shotgun metagenomics on wide scale projects is the high cost associated with the approach. However, we demonstrate that—for some applications—it is possible to use shallow shotgun metagenomics to characterize complex biological matrices while reducing costs. We measured the variation of several summary statistics simulating a decrease in sequencing depth by randomly subsampling a number of reads. The main statistics that were compared are alpha diversity estimates, species abundance, detection threshold, and ability of reconstructing the metagenome in terms of length and completeness. Our results show that a classification of prokaryotic, eukaryotic and viral communities can be accurately performed even using very low number of reads, both in mock communities and in real complex matrices. With samples of 100,000 reads, the alpha diversity estimates were in most cases comparable to those obtained with the full sample, and the estimation of the abundance of all the present species was in excellent agreement with those obtained with the full sample. On the contrary, any task involving the reconstruction of the metagenome performed poorly, even with the largest simulated subsample (1M reads). The length of the reconstructed assembly was smaller than the length obtained with the full dataset, and the proportion of conserved genes that were identified in the meta-genome was drastically reduced compared to the full sample. Shallow shotgun metagenomics can be a useful tool to describe the structure of complex matrices, but it is not adequate to reconstruct—even partially—the metagenome.

The exponential function exp(x) = L' OXn/n! can be defined in any domain in which the infinite series converges; thus, in particular, it makes sense in any Banach algebra. This note is concerned with characterizing the range of exp in Mn(R), the set of all n X n real matrices, which is the simplest nontrivial real Banach algebra. Let us agree to write log B = A whenever exp A = B. Then we wish to determine which real matrices possess a real logarithm. It is elementary that for the domain R the range of exp is R + = { y > 0) and for the domain the range is C {O}. For the domain Mn(C), consisting of all n X n complex matrices, it is not too difficult to show that the range is GLn(C), the set of all nonsingular complex matrices. That it can be no larger than GLn(C) follows from the Jacobi identity det(exp A) = exp(tr A), a classical result which is also a consequence of remarks below. Standard infinite series expansions for the logarithm can be used to construct log B for certain large open sets in GLn(C). For example, if IIB -II < 1, where denotes any operator norm, then a logarithm can be defined by the series