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The stability and inertia of sign pattern matrices with entries in { + , − , 0 } associated with dynamical systems of second-order ordinary differential equations x ¨ = A x ˙ + B x are studied, where A and B are real matrices of order n. An equivalent system of first-order differential equations has coefficient matrix C = [ A B I O ] of order 2n, and eigenvalue properties are considered for sign patterns C = [ A B D O ] of order 2n, where A , B are the sign patterns of A , B respectively, and D is a positive diagonal sign pattern. For given sign patterns A and B where one of them is a negative diagonal sign pattern, results are determined concerning the potential stability and sign stability of C , as well as the refined inertia of C . Applications include the stability of such dynamical systems in which only the signs rather than the magnitudes of entries of the matrices A and B are known.

We prove that semialgebraic sets of rectangular matrices of a fixed rank, of skew-symmetric matrices of a fixed rank and of real symmetric matrices whose eigenvalues have prescribed multiplicities are minimal submanifolds of the space of real matrices of a given size.

Truncated singular value decomposition is a reduced version of the singular value decomposition in which only a few largest singular values are retained. This paper presents a novel perturbation analysis for the truncated singular value decomposition for real matrices. First, we describe perturbation expansions for the singular value truncation of order $r$. We extend perturbation results for the singular subspace decomposition to derive the first-order perturbation expansion of the truncated operator about a matrix with rank greater than or equal to $r$. Observing that the first-order expansion can be greatly simplified when the matrix has exact rank $r$, we further show that the singular value truncation admits a simple second-order perturbation expansion about a rank-$r$ matrix. Second, we introduce the first-known error bound on the linear approximation of the truncated singular value decomposition of a perturbed rank-$r$ matrix. Our bound only depends on the least singular value of the unperturbed matrix and the norm of the perturbation matrix. Intriguingly, while the singular subspaces are known to be extremely sensitive to additive noises, the newly established error bound holds universally for perturbations with arbitrary magnitude. Finally, we demonstrate an application of our results to the analysis of the mean squared error associated with the TSVD-based matrix denoising solution., Comment: Accepted to Linear Algebra and Its Applications

Let $\mathcal{S}_n$ be the set of all $n$-by-$n$ symmetric real matrices, and let $\mathcal{C}_n$ be the copositive cone, that is, the set of all matrices $a\in\mathcal{S}_n$ that fulfill the condition $u^\top a u\geqslant0$ for all $n$-vectors $u$ with nonnegative entries. We prove that a linear mapping $\varphi:\mathcal{S}_n\to \mathcal{S}_n$ satisfies $\varphi(\mathcal{C}_n)=\mathcal{C}_n$ if and only if $$\varphi(x)=m^\top xm$$ for a fixed monomial matrix $m$ with nonnegative entries., Comment: 5 pages

We show that the structured singular value of a real matrix with respect to five full complex uncertainty blocks equals its convex upper bound. This is done by formulating the equality conditions as a feasibility SDP and invoking a result on the existence of a low-rank solution. A counterexample is given for the case of six uncertainty blocks. Known results are also revisited using the proposed approach.

The general problem under consideration is the determination of the best (i.e., smallest) constant C p , q , r such that ‖ X Y − Y X ‖ p ≤ C p , q , r ‖ X ‖ q ‖ Y ‖ r for all n × n matrices X and Y , where ‖ ⋅ ‖ p is the Schatten p-norm. Among the open situations, the problem is solved when (i) 2 p ∞ , q = r = 1 ; (ii) 2 p ∞ , q = 1 , 1 r 2 , and X, Y are 2 × 2 real matrices. The result for (i) is obtained via the study of the region (recently obtained in [5] ) of singular values of the commutator X Y − Y X , where X and Y are normalized rank one matrices. As a result of independent interest, for 2 × 2 real matrices, the region of singular values of the commutator X Y − Y X , where X is rank one and normalized, and ‖ Y ‖ r = 1 , is determined. The result for (ii) is a consequence of the case when r lies between 1 and 2.

Tracing the herbicide bixlozone and its metabolites in food is necessary to assess their risks to human health. In the study, a rapid and effective analytical method using the quick, easy, cheap, effective, rugged, and safe (QuEChERS) method for the simultaneous determination of bixlozone and its metabolites (2,4-dichlorobenzoic acid, 3-hydroxy-propanamide-bixlozone, and 5'-hydroxy-bixlozone) in plant and animal samples (tomato, cucumber, apple, wheat flour, meat, milk, and egg) was developed based on ultra-performance liquid chromatography-tandem mass spectrometry. The method was validated based on the linearity (R2 > 0.99), sensitivity (limit of quantification = 0.01 mg/kg), recovery (70.2%-115.1%), and precision (intra-day 1.2%-17.6%, inter-day 0.3%-16.0%). Detection was achieved within 6.0 min. The method is reliable for the determination of four target compounds in all seven matrices. The satisfactory validation criteria and successful application show that the proposed methodology is suitable for the detection of four target compounds in real matrices. This article is protected by copyright. All rights reserved.

We consider linear Hamiltonian equations in R 2 n of the following type d γ d t ( t ) = J 2 n A ( t ) γ ( t ) , γ ( 0 ) ∈ Sp ( 2 n , R ) , where J = J 2 n = def [ 0 Id n − Id n 0 ] and A : t ↦ A ( t ) is a C 1 curve in the space of 2 n × 2 n real matrices which are symmetric. Then, t ↦ γ ( t ) is a C 2 curve in the space of 2 n × 2 n (real) symplectic matrices. We obtain second order asymptotics for the eigenvalues bifurcated from non-real Krein indefinite eigenvalues with algebraic multiplicity two and geometric multiplicity one. As a corollary, we obtain a simple formula about the derivative of the sum of the bifurcated eigenvalues at time t = 0 . In the end, we discuss possible potential applications for the linear stability of the elliptic Lagrangian solutions of the planar three-body problem.

We investigate the main phase space properties of the QR-flow when restricted to upper Hessenberg matrices. A complete description of the linear behavior of the equilibrium matrices is given. The main result classifies the possible \begin{document}$ \alpha $\end{document} - and \begin{document}$ \omega $\end{document} -limits of the orbits for this system. Furthermore, we characterize the set of initial matrices for which there is convergence towards an equilibrium matrix. Several numerical examples show the different limit behavior of the orbits and illustrate the theory.

A proof of the statement per(A ∘ B) ≤ per(A)per(B) is given for 4 × 4 positive semidefinite real matrices. The proof uses only elementary linear algebra and a rather lengthy series of simple inequalities.

An m × n matrix R with nonnegative entries is called row stochastic if the sum of entries on every row of R is 1. Let Mm,n be the set of all m × n real matrices. For A, B ∈ Mm,n, we say that A is row Hadamard majorized by B (denoted by A ≺ RHB) if there exists an m × n row stochastic matrix R such that A = R ο B, where X ο Y is the Hadamard product (entrywise product) of matrices X, Y ∈ Mm,n. In this paper, we consider the concept of row Hadamard majorization as a relation on Mm,n and characterize the structure of all linear operators T: Mm,n → Mm,n preserving (or strongly preserving) row Hadamard majorization. Also, we find a theoretic graph connection with linear preservers (or strong linear preservers) of row Hadamard majorization, and we give some equivalent conditions for these linear operators on Mn.

Concentration procedures have always been implemented when trace analysis of compounds in real matrices is contemplated. A variety of concentration strategies have been reported aiming at decreasin...

In this paper, we investigate on the number of all possible real matrices representing a quaternion number as three4×4skew-symmetric matrices plus the identity matrix of order 4, and how to determine these matrices. We establish that there are 96 distinct real matrices having this property, and by matrix row operations, we obtain these matrices.

The purpose of this note is to obtain rational criteria for diagonalizability of real matrices through the analysis of the moment and Gram matrices associated to a given real matrix. These concepts were introduced by Horn and Lopatin in [R.A. Horn and A.K. Lopatin. The moment and Gram matrices, distinct eigenvalues and zeroes, and rational criteria for diagonalizability. Linear Algebra and its Applications, 299:153-163, 1999] for complex matrices. However, when the matrix is real, it is possible to combine their results with the Borchardt-Jacobi Theorem, in order to get new and noteworthy rational criteria.

In this work (Part I), we reinvestigate the study of the stability of the Covid-19 mathematical model constructed by Shah et al. (2020) [1]. In their paper, the transmission of the virus under different control strategies is modeled thanks to a generalized SEIR model. This model is characterized by a five dimensional nonlinear dynamical system, where the basic reproduction number can be established by using the next generation matrix method. In this work (Part I), it is established that the disease free equilibrium point is locally as well as globally asymptotically stable when . When , the local and global asymptotic stability of the equilibrium are determined employing the second additive compound matrix approach and the Li-Wang’s (1998) stability criterion for real matrices [2]. In the second paper (Part II), some control parameters with uncertainties will be added to stabilize the five-dimensional Covid-19 system studied here, in order to force the trajectories to go to the equilibria. The stability of the Covid-19 system with these new parameters will also be assessed in Intissar (2020) [3] applying the Li-Wang criterion and compound matrices theory. All sophisticated technical calculations including those in part I will be provided in appendices of the part II.

In this paper, we study nonlinear matrix equations $$\begin{aligned} X^p=A+\sum \limits _{i=1}^m M_i^T(X\#B)M_i \end{aligned}$$ and $$\begin{aligned} X^p=A+\sum \limits _{i=1}^j M_i^T(X\#B)M_i+\sum \limits _{i=j+1}^m M_i^T(X^{-1}\#B)M_i, \end{aligned}$$ where p, m, j are positive integers, $$1\le j\le m$$ , A, B are $$n\times n$$ positive definite matrices and $$M_i(i=1,2,3,\ldots ,m)$$ are $$n\times n$$ nonsingular real matrices. Based on some fixed point theorems for monotone and mixed monotone operators in ordered Banach spaces and some properties of cone, we prove that these equations always have a unique positive definite solution. In addition, an iterative sequence can be given to approximate the unique positive definite solution by employing a multi-step stationary iterative method.

The minimum rank of a simple graph G is defined to be the smallest possible rank over all symmetric real matrices whose i j t h entry (for i ≠ j ) is nonzero whenever ( i , j ) is an edge in G and ...

We study the differential-geometric properties of the loci of fixed points of the elliptic isometries of the manifold of positive definite real matrices with the trace metric. We also give an explicit description of such loci and in particular we find their De Rham decomposition.

For a family of real matrices with entries polynomially depending on parameters, symbolic algorithms are proposed for verification of the Routh – Hurwitz stability under parameter variations in a g...

Let $$\mathbf{A}= (A_1, \ldots , A_m)$$ , where $$A_1, \ldots , A_m$$ are $$n\times n$$ real matrices. The real joint (p, q)-matricial range of $$\mathbf{A}$$ , $${\varLambda }^{{\mathbb {R}}}_{p,q}(\mathbf{A})$$ , is the set of m-tuple of $$q\times q$$ real matrices $$(B_1, \ldots , B_m)$$ such that $$(X^*A_1X, \ldots , X^*A_mX) = (I_p\otimes B_1, \ldots , I_p \otimes B_m)$$ for some real $$n \times pq$$ matrix X satisfying $$X^*X = I_{pq}$$ . It is shown that if n is sufficiently large, then the set $${\varLambda }^{{\mathbb {R}}}_{p,q}(\mathbf{A})$$ is non-empty and star-shaped. The result is extended to bounded linear operators acting on a real Hilbert space $${{{\mathcal {H}}}}$$ , and used to show that the joint essential (p, q)-matricial range of $$\mathbf{A}$$ is always compact, convex, and non-empty. Similar results for the joint congruence matricial ranges on complex operators are also obtained.

We regard the real symplectic group $$Sp(2n,{{\mathbb {R}}})$$ as a constraint submanifold of the $$2n\times 2n$$ real matrices $${\mathcal {M}}_{2n}({{\mathbb {R}}})$$ endowed with the Euclidean (Frobenius) metric, respectively as a submanifold of the general linear group $$Gl(2n,{{\mathbb {R}}})$$ endowed with the (left) invariant metric. For a cost function that defines an optimization problem on the real symplectic group we give a necessary and sufficient condition for critical points and we apply this condition to the particular case of a least square cost function. In order to characterize the critical points we give a formula for the Hessian of a cost function defined on the real symplectic group, with respect to both considered metrics. For a generalized Brockett cost function we present a necessary condition and a sufficient condition for local minimum. We construct a retraction map that allows us to detail the steepest descent and embedded Newton algorithms for solving an optimization problem on the real symplectic group.

System identification methods (SIMs) are powerful tools for structural health monitoring. SIMs are used either for identification of modal parameters or for extraction of structural characteristic matrices. The current study aims to identify the real matrices of a structure using structural dynamics theory and stochastic subspace identification based on realization theory. The study focuses on extraction of the condensed mass and stiffness matrices of shear buildings by means of ambient excitation responses for limited structural degrees of freedom. Realization theory states that there are minimal realizations for appropriate identification of the main real-system matrices. The present study shows that, by using values smaller than the minimal realization, condensed structural matrices can be correctly identified. This is accomplished by accurate estimation of the full real-system order. It is shown that successive repetitions of this approach can lead to complete structural matrices. The practical application of this procedure is examined using an experimental model and an analytical six-story shear model. Analysis revealed that, even in the presence of noise, this method can accurately identify structural matrices in all cases with high precision.

In this paper, we establish some refinements of the following triangle inequality proved by T.-Y. Tam: $$ {{\widetilde{\sigma }}} (x + y) \prec _{mt} {{\widetilde{\sigma }}} (x) + {{\widetilde{\sigma }}} (y) $$ for real matrices x and y with the modified singular value operator $$ {{\widetilde{\sigma }}} (\cdot ) $$ and Miranda-Thompson’s majorization $$ \prec _{mt} $$ on $$ {\mathbb {R}}^{n}$$ related to the group of special orthogonal matrices.

We obtain a sufficient condition for the convexity of quaternionic numerical range for complex matrices in terms of its complex numerical range. It is also shown that the Bild coincides with complex numerical range for real matrices. From this result we derive that all real matrices have convex quaternionic numerical range. As an example we fully characterize the quaternionic numerical range of 2 × 2 real matrices.

Synchronization is studied in an array of identical oscillators undergoing small vibrations. The overall coupling is described by a pair of matrix-weighted Laplacian matrices; one representing the dissipative, the other the restorative connectors. A construction is proposed to combine these two real matrices in a single complex matrix. It is shown that whether the oscillators synchronize in the steady state or not depend on the number of eigenvalues of this complex matrix on the imaginary axis. Certain refinements of this condition for the special cases, where the restorative coupling is either weak or absent, are also presented.

In this contribution, the features and the advantages of the modified hollow fibre microextraction (HFµE) technique are discussed, by using two membranes (dual-HFµE) per sample, loaded with conveni...

Let $${\mathbf {M}}_{n,m}$$ be the set of all n-by-m real matrices and let the relation $$\prec $$ on $${\mathbf {M}}_{n,m}$$ be the multivariate majorization. A linear mapping $$\Phi \!:{\mathbf {M}}_{n,m}\longrightarrow {\mathbf {M}}_{n,m}$$ is said to be a linear preserver of multivariate majorization if $$\Phi (X)\prec \Phi (Y)$$ whenever $$X\prec Y$$. Also, $$\Phi $$ is said to be a local linear preserver of multivariate majorization if for each $$X\in {\mathbf {M}}_{n,m}$$ there exists a linear mapping $$\Phi _X$$ on $${\mathbf {M}}_{n,m}$$ that preserves, multivariate majorization and $$\Phi (X)=\Phi _X(X)$$. In this paper we introduce two types of strong local linear preservers of multivariate majorization on $${\mathbf {M}}_{n,m}$$. Then we find a relation between these types of strong local linear preservers and finally we characterize the structure of these linear maps.

The notion of inverse M-matrices has been quite well studied and many of their properties have been documented in the literature. In this article, a proposal to consider the larger class of inverse H-matrices, is made. First, certain special types of real matrices are identified as inverse H-matrices. While these matrices are not inverse M-matrices in general, their definitions are inspired by matrices that are known to be inverse M-matrices. In the rest of the article, motivated by results that hold for inverse M-matrices, the endeavor will be to attempt at proving corresponding results for inverse H-matrices. It is shown that for the full class of inverse H-matrices, many of these extensions are false. Nevertheless, two subclasses of real inverse H-matrices are identified, where these generalizations are shown to hold.

In this work, we propose an innovative high throughput (HT) apparatus using the bar adsorptive microextraction (BAμE) technique, which enables the simultaneous enrichment of up to 100 samples. This novel configuration was combined with microliquid desorption and high-performance liquid chromatography-diode array detection to monitor trace levels of eight benzodiazepines (diazepam, prazepam, bromazepam, oxazepam, lorazepam, alprazolam, temazepam and loflazepate) in biological samples. The proposed methodology was fully developed, optimized and validated, resulting in suitable intraday and interday precision (RSD ≤ 15%), with recovery yields ranging from 33.0% to 104.5%. The lower limits of quantification were between 20.0 and 100.0 µg L−1, using 1.0 mL of urine and 0.5 mL of plasma or serum samples. The application of the proposed methodology to real matrices resulted in average sample preparation time of around 2 min per sample, demonstrating that it is user-friendly, cost-effective and a rapid decision-making tool, whenever large number of samples are involved.

Quadratic pencils λ2M + λC + K, where M, C, and K are n × n real matrices, arise in a broad range of important applications. Its spectral properties affect the vibration behavior of the underlying system which often consists of many elements coupled together through an intricate network of inter-connectivities. It is known that an n-degree-of-freedom system with semi-simple eigenvalues can be reduced to, without tampering with the innate vibration properties, n mutually independent single-degree-of-freedom subsystems, referred to as total decoupling. This paper revisits the problem with the additional constraint that the masses should stay invariant throughout the reduction process. Rescaling the masses if necessary, M is assumed to be the identity matrix. Isospectral flows are derived to either totally or partially decouple C and K to independent units of modules. Indeed, the same framework can be tailored to handle any kinds of desired structure. Two new results are obtained. First, the global convergence is guaranteed by using the Łojasiewicz gradient inequality. Second, bounds on errors due to numerical integration and floating-point arithmetic calculation are derived, which can be used for assessing the quality of the transformation. Numerical experiments on four distinct scenarios are given to demonstrate the capabilities of the framework of handling the decoupling problem.

In the present study, we have developed a novel electroanalytical method based on coupling of four-way multivariate calibration (MVC) models with third-order differential pulse voltammetric (DPV) data for simultaneous determination of amaranth (AM), tartrazine (TT) and quinoline yellow (QY) in the presence of sunset yellow (SY) as uncalibrated interference at a glassy carbon electrode (GCE). The third-order DPV data were recorded by changing pulse height and pulse time as instrumental parameters. After performing some preprocessing actions such as baseline correction by asymmetric least squares spline regression (AsLSSR) and potential shift correction by correlation optimized warping (COW) on the raw data, two well-known third-order algorithms including unfolded partial least squares/residual trilinearization (U-PLS/RTL) and multidimensional partial least squares/RTL (N-PLS/RTL) were used to build four-way calibration models and their performance was checked by predicting the concentrations of the validation and test sets as synthetic samples. The results confirmed more superiority of N-PLS/RTL than U-PLS/RTL in synthetic samples which encouraged us to apply the N-PLS/RTL to the analysis of real samples. The results of the analysis of real samples showed that the N-PLS/RTL had an acceptable potential for predicting concentrations of AM, TT and QY in real matrices. Fil: Ghanbari, Kazhal. Kermanshah University Of Medical Sciences; Irán Fil: Roushani, Mahmoud. Kermanshah University Of Medical Sciences; Irán Fil: Farzadfar, Farshad. Kermanshah University Of Medical Sciences; Irán Fil: Goicoechea, Hector Casimiro. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional del Litoral; Argentina Fil: Jalalvand, Ali R.. Kermanshah University Of Medical Sciences; Irán

We construct samples of real matrices with number of rows greater than the number of columns and satisfying four requirements: the squares of the rows equal one, the squares of the columns equal each other, the columns are pairwise orthogonal, the sum of the components of each column is zero, except for two cases. In the first case, the number of rows is odd and the number of columns is one. In the second case, the number of rows is odd and the number of columns is two less than the number of rows. It is proved that in these cases, there are no matrices satisfying the four specified requirements. The place of matrices satisfying the four specified requirements is shown in the theory of errors in measuring systems such as GPS.

In this paper, we consider the quaternion Stein equation. By using a new kind of real representation of quaternion matrix, the semi-tensor product of matrices and the Moore-Penrose generalized inverse, we obtain the expressions of the minimal norm least squares tridiagonal (Anti-) Hermitian solution of quaternion Stein equation. Our resulting expression is expressed only by real matrices, and the algorithm only includes real operations. We provide numerical algorithms and present two numerical examples to illustrate the efficiency of our method.

In this paper, we completely characterize the linear maps $$\phi :\mathcal {M} \rightarrow \mathcal {M}$$ that preserve the Lorentz-cone spectrum, when $$\mathcal {M}$$ is one of the following subspaces of the space $$M_{n}$$ of $$n\times n$$ real matrices: the subspace of diagonal matrices, the subspace of block-diagonal matrices $$\widetilde{A}\oplus [a]$$ , where $$\widetilde{A}\in M_{n-1}$$ is symmetric, and the subspace of block-diagonal matrices $$\widetilde{A}\oplus [a]$$ , where $$\widetilde{A}\in M_{n-1}$$ is a generic matrix. In particular, we show that $$\phi$$ should be what we call a standard map, namely, a map of the form $$\phi (A)=PAQ$$ for all $$A\in \mathcal {M}$$ or $$\phi (A)=PA^{T}Q$$ for all $$A\in \mathcal {M},$$ for some matrices $$P,Q\in M_{n}$$ . We then characterize the standard maps preserving the Lorentz-cone spectrum, when $$\mathcal {M}$$ is the subspace $$S_{n}$$ of symmetric matrices. The case $$\mathcal {M=}M_{n}$$ was considered in a recent paper by Seeger (LAA 2020). We include it here for completeness.

We say that a square real matrix $M$ is \emph{off-diagonal nonnegative} if and only if all entries outside its diagonal are nonnegative real numbers. In this note we show that for any off-diagonal nonnegative symmetric matrix $M$, there exists a nonnegative symmetric matrix $\widehat{M}$ which is sparse and close in spectrum to $M$., Comment: 9 pages

As was already mentioned in the Chapter 1, complex numbers may appear as matrix elements. Moreover, the characteristics of real matrices (such as eigenvalues, see Chap. “Vector Spaces” on page 226) in some cases appear to be complex. In this connection, let us discuss the methods of algebra of complex numbers.

We characterize the "structures" defining a class of real matrices—called nests—given by A in ℝn×n such that, up to relabeling, all of its leading principal submatrices indexed by {1, 2,⋯, k} satisfy sgn det A[{1, 2,⋯k}] = (−1)k k ≤ n By structure we elements mean of {−1, 0, 1}n×n, which circumscribe the signs of realizations within a class of real matrices. We frame the problem of characterizing structures in a form amenable to classic control theory, and that generalizes to any class of matrices defined by potentially Hurwitz structures, such as nests.

In this paper we study the endpoint geodesics problem on the differentiable manifold of real positive definite matrices. The objective is to connect two points given as initial data on that space by a unique geodesic. We first recall partially well-known facts about the differential geometry of this manifold. Then we consider further features, namely the property of being an extrinsic symmetric space together with associated Riemannian isometries. This paper is essentially self-contained and therefore accessible to a wide audience.

Let Ln(d)≤ be the set of sequences of real matrices L=(L1,…,Ln), Li∈Rp×m, i=1,…,n, admitting a minimal partial realisation of order d and such that the sum of their largest partial row and column K...

Each connected component of the Gershgorin circles of a matrix contains exactly as many eigenvalues as circles are involved. Thus the power set product of all circles is an inclusion of the determinant if all circles are disjoint. We prove that statement to be true for real matrices, even if their circles overlap.

Each connected component of the Gershgorin circles of a matrix contains exactly as many eigenvalues as circles are involved. Thus, the Minkowski (set) product of all circles contains the determinant if all circles are disjoint. In [S.M. Rump. Bounds for the determinant by Gershgorin circles. Linear Algebra and its Applications, 563:215--219, 2019.], it was proved that statement to be true for real matrices whose circles need not to be disjoint. Moreover, it was asked whether the statement remains true for complex matrices. This note answers that in the affirmative. As a by-product, a parameterization of the outer loop of a Cartesian oval without case distinction is derived.

Using the concept of a transposition anti-involution in Clifford algebra $$C \, \ell _{1,1}$$ and the isomorphisms $${\mathbb {H}}_s \cong C \, \ell _{1,1} \cong \text {Mat}(2,{\mathbb {R}}),$$ where $${\mathbb {H}}_s$$ is the algebra of split quaternions and $$\text {Mat}(2,{\mathbb {R}})$$ is the algebra of $$2 \times 2$$ real matrices, one can find the Moore–Penrose inverse $$q^{+}$$ of a non-zero non-invertible split quaternion q. In particular, using a well-known algorithm for finding the Moore-Penrose inverse $$Q^{+}$$ of a non-zero $$2 \times 2$$ matrix Q of rank 1, one can give four governing equations that the (unique) split quaternion $$q^{+}$$ corresponding to $$Q^{+}$$ must satisfy. We show how a dyadic expansion and a Singular Value Decomposition can be found for any split quaternion q, and we relate them to $$q^{+}$$ . Results presented in this paper may be useful in a plethora of recent applications of split quaternions.

In this paper, we propose an efficient method for special least squares solution of the complex matrix equation ( A X B , C X D ) = ( E , F ) . By using the real representation matrices of complex matrices, the particular structure of the real representation matrices, the Moore–Penrose generalized inverse and the Kronecker product, we obtain the explicit expression of the minimal norm least squares Hermitian solution of the complex matrix equation ( A X B , C X D ) = ( E , F ) , which was studied by a product of matrices and vectors in Wang et al. (2016). Our resulting formulas only involve real matrices, and the corresponding algorithm only performs real arithmetic. Therefore our proposed method is more effective and portable. Finally, we give three numerical examples to illustrate the effectiveness of our proposed method.

In this paper, development and optimization of a new method for determination of the three hormones (estrone, 17-β-estradiol and 17-α-ethinylestradiol) of the Watch list were described. The validated method relies on isotope dilution—two-step solid phase extraction–liquid phase chromatography mass spectrometry method. The measurement procedure validation has been performed in real matrices, including matrix with high suspended particulate matter and high organic carbon contents, to demonstrate its fitness for purpose. Method performances were in agreement with the requirements of the decision 2015/495/EU (maximal acceptable method detection limit and whole-water analysis). Limits of quantification of the method were of 0.4 ng L−1 for estrone and 17-β-estradiol and 0.1 ng L−1 for 17-α-ethinylestradiol. Expanded uncertainties (k = 2) at limit of quantification were equal to 35 % for estrone and 17-β-estradiol and 50 % for 17-α-ethinylestradiol. The method was successfully implemented to monitor French inland surface waters contamination. The survey reveals a chronic state of contamination by estrone (96 % quantification rate) and significant one by 17-β-estradiol and 17-α-ethinylestradiol (15 % quantification rate). Maximum measured concentrations are above predicted no-effect concentrations, indicating potential risk toward the environment.

Here, we report on the design, fabrication, and characterization of a prototype of a portable detection system, able to determine if the contamination level of Ochratoxin A (OTA) is below or above the limit imposed by the EU Commission for a specific food. The operating principle of the system is the real-time monitoring of a chromatographic run when an extract of the sample under analysis is spotted on a high-performance thin-layer chromatographic plate. The idea is to induce naturally the fluorescence of the OTA molecules by using an ultraviolet excitation source and to detect the re-emitted light with an array of amorphous silicon photosensors, whose photocurrents will be proportional to the amount of OTA molecules. The system performances have been verified on sample containing pure OTA molecule and OTA extracted from fortified real matrices of wine and beer. The results showed a minimum detectable quantity of OTA of 0.2 ng and repeatability better than 5%. The analysis of the sample is made simultaneously with a reference, which, in the proposed system, corresponds to the amount of OTA equal to the limit for the specific food commodities. Taking into account the extraction efficiency and the minimum acceptable presence of OTA in such food commodities, we infer that the minimum amount of real matrix to be investigated is as low as 5.5 mL.

Let $$\mathbf M _{m,n}$$ be the set of all $$m\times n$$ real matrices. A matrix $$A\in \mathbf M _{m,n}$$ is said to be a dense matrix if there are no zeros between two nonzero entries for every line (row or column) of this matrix. In this paper, we find the structure of linear maps $$T:\mathbf M _{m,n} \rightarrow \mathbf M _{m,n}$$ that strongly preserve dense matrices, i.e., T(A) is a dense matrix if and only if A is a dense matrix.

Let Xn = (xij) be a k×n data matrix with complex‐valued, independent and standardized entries satisfying a Lindeberg‐type moment condition. We consider simultaneously R sample covariance matrices Bnr=1nQrXnXn∗Qr⊤,1≤r≤R, where the Qr's are non‐random real matrices with common dimensions p×k(k≥p). Assuming that both the dimension p and the sample size n grow to infinity, the limiting distributions of the eigenvalues of the matrices {Bnr} are identified, and as the main result of the paper, we establish a joint central limit theorem (CLT) for linear spectral statistics of the R matrices {Bnr}. Next, this new CLT is applied to the problem of testing a high‐dimensional white noise in time series modelling. In experiments, the derived test has a controlled size and is significantly faster than the classical permutation test, although it does have lower power. This application highlights the necessity of such joint CLT in the presence of several dependent sample covariance matrices. In contrast, all the existing works on CLT for linear spectral statistics of large sample covariance matrices deal with a single sample covariance matrix (R = 1).

Thompson obtained necessary and sufficient conditions on the existence of a real square matrix with prescribed diagonal elements, prescribed singular values and non-negative determinant. Th...

The vertices of the commuting graph of a semigroup S are the noncentral elements of this semigroup, and its edges join all pairs of elements g, h that satisfy the relation gh = hg. The paper presents a proof of the fact that the diameter of the commuting graph of the semigroup of real matrices of order n ≥ 3 is equal to 4. A survey of results in that subject matter is presented, and several open problems are formulated.