Null Spaces Research Articles

Strong B-T inverse

New-typed matrix inverses based on the Hartwig-Spindelböck decomposition were investigated, which is called the strong B-T inverse, as a generalization of the B-T inverse. The relationships between the above inverse and other matrix inverses were established. Several sufficient and necessary conditions of the strong B-T inverse are obtained via the column and null spaces.

Arrive and retreat: London’s ultra-prime apartment blocks as dark urban renaissance

The expansion of an archipelago of residential blocks across London’s West End has created an urban renaissance of sorts, but it is a rebirth that has targeted the expanded ranks of the global super-rich, yielding a visibly luxurious, often vacant residential remaking, rather than the aspirations for a diverse, vibrant and open urbanism envisaged some years ago for the city. New luxury apartment blocks offer high levels of privacy, security and the seamless integration of bodies and vehicles into underground or enclosed entry points. These ‘ultraland’ blocks are less places of conviviality and encounter, offering instead a kind of dark space urbanism that facilitates a partial social engagement with local environs by barely present owners. Combining an analysis of the planning applications for these developments with extensive street observation in London’s West End we discuss the social and spatial integration of these developments. We argue that a key effect is one of social ‘reduction’—the sense of null spaces that absorb bodies, furnishings and cars in ways that are the antithesis of earlier ambitions for socially vital and democratic urban spaces. We discuss the significance of these changes for contemporary life in the city more broadly.

A rank-updating technique for the Kronecker canonical form of singular pencils

For a linear time-invariant system x˙(t)=Ax(t)+Bu(t), the Kronecker canonical form (KCF) of the matrix pencil (sI−A|B) provides the controllability indices, also called column minimal indices, of the system and their sum corresponds to the dimension of the controllable subspace. In this paper we introduce a fast numerical algorithm for computing the sets of column/row minimal indices of a singular pencil sF−G using a rank-updating technique and the properties of piecewise arithmetic progression sequences defined by the size of the null spaces of appropriate Toeplitz matrices. The method is demonstrated and tested on various data sets.

Assessing large-scale mantle compositional heterogeneity from machine learning analysis of 28 global P- and S-wave tomography models

SUMMARY Differences between P- and S-wave models have been frequently used as evidence for the presence of large-scale compositional heterogeneity in the Earth's mantle. Our two-step machine learning (ML) analysis of 28 P- and S-wave global tomographic models reveals that, on a global scale, such differences are for the most part not intrinsic and could be reduced by changing the models in their respective null spaces. In other words, P- and S-wave images of mantle structure are not necessarily distinct from each other. Thus, a purely thermal explanation for large-scale seismic structure is sufficient at present; significant mantle compositional heterogeneities do not need to be invoked. We analyse 28 widely used tomographic models based on various theoretical approximations ranging from ray theory (e.g. UU-P07 and MIT-P08), Born scattering (e.g. DETOX) and full-waveform techniques (e.g. CSEM and GLAD). We apply Varimax principal component analysis to reduce tomography model dimensionality by 83 percent, while preserving relevant information (94 percent of the original variance), followed by hierarchical clustering (HC) analysis using Ward's method to quantitatively categorize all models into hierarchical groups based on similarities. We found two main tomography model clusters: Cluster 1, which we called ‘Pure P wave’, is composed of six P-wave models that only use longitudinal body wave phases (e.g. P, PP and Pdiff); and Cluster 2, which we called ‘Mixed’, includes both P- and S-wave models. P-wave models in the ‘Mixed’ cluster use inversion methods that include inputs from other geophysical and geological data sources, and this causes them to be more similar to S-wave models than Pure P-wave models without significant loss of fitness to P-wave data. Given that inclusion of new data classes and seismic phases in more recent tomographic models significantly changes imaged seismic structure, our ML assessment of global tomography model similarity may improve selection of appropriate P- and S-wave models for future global tomography comparative studies.

Spectral properties of the Rhaly operator on weighted null sequence spaces and associated operator ideals

In this paper, a comprehensive study is made on the continuity, compactness, and spectrum of the lower triangular terraced matrix, introduced by Rhaly [[Formula: see text]-Cesaro matrices, Houston J. Math. 15(1) (1989) 137–46], acting on the weighted null sequence spaces with bounded, strictly positive weights. Several spectral subdivisions such as point spectrum, residual spectrum, and continuous spectrum are also discussed. In addition, a new class of operator ideal [Formula: see text] associated to the Rhaly operator on weighted [Formula: see text] space is defined using the concept of [Formula: see text]-number and it is proved that under certain condition, [Formula: see text] forms a quasi-Banach closed operator ideal.

Simplified form-finding for tensegrity structures through reference joints of symmetry orbits

Tensegrity structures are a class of free-standing and prestressed cable-strut structures. Different from traditional elastic structures, both geometric shapes and initial prestress distributions of tensegrity systems are indeterminate. Thus, it is of great significance to develop a robust and effective form-finding method for tensegrity systems. Based on reference joints of symmetry orbits and force density theory, this study proposes a simplified form-finding method for tensegrity structures. This method makes full use of the inherent symmetry of structural systems. It introduces a series of orthogonal transformation matrices for certain reference joints to represent independent force equilibriums of similar joints from the same symmetry orbit. Then, the original form-finding problem is simplified into finding null spaces for several small-sized force density matrices about the reference joints. Thus, the force density of each type of components and the nodal coordinate of each joint can be effectively obtained. Illustrative examples show that the proposed method can significantly simplify the computational complexity of form-finding for tensegrity systems with symmetry orbits.

Downlink Massive MU-MIMO With Successively-Regularized Zero Forcing Precoding

In this letter, we consider linear precoding for downlink massive multi-user (MU) multiple-input multiple-output (MIMO) systems. We propose the novel successively-regularized zero forcing (SRZF) precoding, which exploits successive null spaces of the MIMO channels of the users, along with regularization, to control the inter-user interference (IUI) and to enhance performance and robustness to imperfect channel state information (CSI) at the base station (BS). We study the IUI characteristics of the proposed SRZF precoding for perfect and imperfect CSI at the BS. Furthermore, via computer simulations, we compare the sum rate of SRZF precoding with those of several baseline schemes including conventional and regularized zero forcing (ZF) precoding. Our simulation results reveal that, for massive MIMO systems with inter-user channel correlations, the proposed SRZF precoding significantly outperforms the considered baseline schemes for both perfect and imperfect CSI at the BS.

Bilateral Fast Low-Rank Representation With Equivalent Transformation for Subspace Clustering

In recent years, low-rank representation (LRR) has received increasing attention on subspace clustering. Due to inevitable matrix inversion and singular value decomposition in each iteration, however, most of existing LRR algorithms may suffer from high computational complexity, and hence can not cope with the large-scale sample data commendably. To overcome this problem, in this paper, we propose a bilateral fast low-rank representation (BFLRR), which has a linear time complexity with respect to the number of samples. Specifically, we introduce the equivalent transformation method to remove the null spaces of both the columns and rows of the coefficient matrix so that a hypercompact coefficient matrix can be learned. Furthermore, the proposed BFLRR is embedded into a distributed framework as DFC-BFLRR to make it more efficient, which utilizes a combination of the global and local projection matrices. Extensive experiments are carried out on real datasets, and the results testify that the proposed methods not only perform faster-computing speed but also obtain favorable clustering accuracy in comparison with the competing methods among large-scale sample data.

Eigenstructure through Matrix Adjugates and Admissible Pairs

The traditional eigenstructure assignment method is reconsidered in this paper, shedding light on its nature and some of its properties. The approach is through matrix adjugates which enable a more eloquent decomposition compared to the lumped solution usually obtained through matrix null spaces. Based on the new approach, the admissible pair (w,z) is formalized. The new approach enables an alternative methodology to the determination of the permissible closed loop eigenvector subspaces and the might be termed the companion input-subspace. Such approach renders the method a formulae-based method as w and z are now obtained formula-wise as opposed to being extracted out of a lumped solution. Compared to the traditional method, the study reveals information such as w and z are independently and explicitly determined. Moreover, newly assigned eigenvalues always result in z≠0, the input matrix B does not influence z and a fundamental feature of the open loop characteristic polynomial is exposed. Furthermore, closed loop eigenvectors associated with repeated eigenvalues can be explicitly computed by means of differentiation as opposed to null space determination. A special form of system representation is highlighted, which considerably eases the calculations. The myriad concepts pointed out have been demonstrated and authenticated through carefully selected examples involving real, complex, repeated eigenvalues, and two practical systems of an electrical nature.

English

In this paper, we investigate properties of  with closed range satisfying the operator equations   In particular, we investigate the invertibility of   with closed range where the Moore-Penrose inverse of T turns out to be the usual inverse of T under some classes of operators. We also deduce the Moore-Penrose inverse of a perturbed linear operator  with closed range where  such that  has closed ranges and  satisfying some given conditions. The relation between the ranges and null spaces of these operators is also shown. Key words: Moore-Penrose inverse, perturbed linear operator, invertibility of operators.

Determinantal Ideals and the Canonical Commutation Relations: Classically or Quantized

We construct homomorphic images of $$su(n,n)^{{\mathbb {C}}}$$ in Weyl Algebras $${{\mathcal {H}}}_{2nr}$$ . More precisely, and using the Bernstein filtration of $${{\mathcal {H}}}_{2nr}$$ , $$su(n,n)^{{\mathbb {C}}}$$ is mapped into degree 2 elements with the negative non-compact root spaces being mapped into second order creation operators. Using the Fock representation of $${{\mathcal {H}}}_{2nr}$$ , these homomorphisms give all unitary highest weight representations of $$su(n,n)^{{\mathbb {C}}}$$ thus reconstructing the Kashiwara–Vergne List for the Segal–Shale–Weil representation. Using an idea from the derivation of the their list, we construct a homomorphism of $$u(r)^{{\mathbb {C}}}$$ into $${{\mathcal {H}}}_{2nr}$$ whose image commutes with the image of $$su(n,n)^{{\mathbb {C}}}$$ , and vice versa. This gives the multiplicities. The construction also gives an easy proof that the ideal of $$(r+1)\times (r+1)$$ minors is prime. Here, of course, $$r\le n-1$$ and for a fixed such r, the space of any irreducible representation of $$su(n,n)^{{\mathbb {C}}}$$ is annihilated by this ideal. As a consequence, these representations can be realized in spaces of solutions to Maxwell type equations. We actually go one step further and determine exactly for which representations from our list there is a non-trivial homomorphism between generalized Verma modules, thereby revealing, by duality, exactly which covariant differential operators have unitary null spaces. We prove the analogous results for $${{\mathcal {U}}}_q(su(n,n)^{{\mathbb {C}}})$$ . The Weyl Algebras are replaced by the Hayashi–Weyl Algebras $${{\mathcal {H}}}{{\mathcal {W}}}_{2nr}$$ and the Fock space by a q-Fock space. Further, determinants are replaced by q-determinants, and a homomorphism of $${{\mathcal {U}}}_q(u(r)^{{\mathbb {C}}})$$ into $${{\mathcal {H}}}{{\mathcal {W}}}_{2nr}$$ is constructed with analogous properties. For this purpose a Drinfeld Double is used. We mention one difference: The quantized negative non-compact root spaces, while still of degree 2, are no longer given entirely by second order creation operators.

On the Poisson equation in exterior domains

We construct a solution of the Poisson equation in exterior domains $\Omega \subset \mathbb R^n,\;n \ge 2,$ in homogeneous Lebesgue spaces $L^{2,q}(\Omega),;1 < q <\infty,$ with methods of potential theory and integral equations. We investigate the corresponding null spaces and prove that its dimensions is equal to $n+1$ independent of $q$.

Joint estimation of biogeochemical model parameters from multiple experiments: A bayesian approach applied to mercury methylation

To characterize complex biogeochemical systems, results from multiple experiments, where each targets a specific subprocess, are commonly combined. The resulting datasets are interpreted through the calibration of biogeochemical models for process inference and predictions. Commonly used calibration approaches of fitting datasets from individual experiments to subprocess models one at a time is prone to missing information shared between datasets and incomplete uncertainty propagation. We propose a Bayesian joint-fitting scheme addressing the above-mentioned concerns by jointly fitting all the available datasets, thus calibrating the entire biogeochemical model in one go using Markov Chain Monte Carlo (MCMC). The identification of null spaces in the parameter distributions from MCMC guided the simplification of certain subprocess models. For example, fast kinetic sorption was replaced by equilibrium sorption, and Monod demethylation was replaced by first-order demethylation. Joint fitting of datasets resulted in complete uncertainty propagation with parameter estimates informed by all available data.

Effect on null spaces of list-mode imaging systems due to increasing the number of attributes.

There are two types of uncertainty in image reconstructions from list-mode data: statistical and deterministic. One source of statistical uncertainty is the finite number of attributes of the detected particles, which are sampled from a probability distribution on the attribute space. A deterministic source of uncertainty is the effect that null functions of the imaging operator have on reconstructed pixel or voxel values. Quantifying the reduction in this deterministic source of uncertainty when more attributes are measured for each detected particle is the subject of this work. Specifically, upper bounds on an error metric are derived to quantify the error introduced in the reconstruction by the presence of null functions, and these upper bounds are shown to be reduced when the number of attributes is increased. These bounds are illustrated with an example of a two-dimensional single photon emission computed tomography (SPECT) system where the depth of interaction in the scintillation crystal is added to the attribute vector.

Cluster consensus on matrix-weighted switching networks

This paper examines cluster consensus for multi-agent systems on matrix-weighted switching networks. Necessary and/or sufficient conditions under which cluster consensus can be achieved are obtained, as well as quantitative characterization of the steady-state of the cluster consensus. Specifically, when the underlying network switches amongst a finite number of networks, a necessary condition for cluster consensus on matrix-weighted switching networks is derived; moreover, it is shown that the steady-state of the nodes lies at the intersection of the null spaces of the matrix-valued Laplacians of the corresponding switching networks. Furthermore, when the underlying network switches amongst an infinite number of networks, the matrix-weighted integral network is employed to provide sufficient conditions for cluster consensus; analogous to the previous case, the quantitative characterization of the corresponding steady-state of the nodes pertains to the null space analysis of matrix-valued Laplacian of the “integral” network. Lastly, conditions for bipartite consensus of matrix-weighted switching networks are provided. Simulation examples demonstrate the presented theoretical results.

Characterizing bipartite consensus on signed matrix-weighted networks via balancing set

In contrast with the scalar-weighted networks, where bipartite consensus can be achieved if and only if the underlying signed network is structurally balanced, the structural balance property is no longer a graph-theoretic equivalence to the bipartite consensus in the case of signed matrix-weighted networks. To re-establish the relationship between the network structure and the bipartite consensus solution, the non-trivial balancing set is introduced which is a set of edges whose sign negation can transform a structurally imbalanced network into a structurally balanced one and the weight matrices associated with edges in this set have a non-trivial intersection of null spaces. We show that necessary and/or sufficient conditions for bipartite consensus on matrix-weighted networks can be characterized by the uniqueness of the non-trivial balancing set, while the contribution of the associated non-trivial intersection of null spaces to the steady-state of the matrix-weighted network is examined. Moreover, for matrix-weighted networks with a positive–negative spanning tree, necessary and sufficient condition for bipartite consensus using the non-trivial balancing set is obtained. Simulation examples are provided to demonstrate the theoretical results.

Sensitivity analysis for sectional stiffness of anisotropic beams: The direct and adjoint methods

This paper concerns with the determination of sectional stiffness matrix and its sensitivity for anisotropic beams with initial curvatures. The governing equations of a three-dimensional beam is formulated to a Hamiltonian system, and the reduction of the Hamiltonian matrix leads to a set of singular, linear equations of the sectional warping and compliance matrix. The left and right null spaces of the singular linear system are explicitly constructed, and the existence and uniqueness of sectional warping and compliance matrix are justified. The direct and adjoint methods for sensitivity analysis of sectional stiffness matrix are developed. The existence and uniqueness of solutions for singular equations resulting from the direct and adjoint methods are proved formally. Numerical examples are presented to validate the proposed methods. The predictions of sensitivities from the direct and adjoint methods are shown to agree well with results of the complex-step method.

Using subspaces of weight matrix for evaluating generative adversarial networks with Fréchet distance

SummaryFréchet inception distance (FID) has gained a better reputation as an evaluation metric for generative adversarial networks (GANs). However, it is subjected to fluctuation, namely, the same GAN model, when trained at different times can have different FID scores, due to the randomness of the weight matrices in the networks, stochastic gradient descent, and the embedded distribution (activation outputs at a hidden layer). In calculating the FIDs, embedded distribution plays the key role and it is not a trivial question from where obtaining it since it contributes to the fluctuation also. In this article, I showed that embedded distribution can be obtained from three different subspaces of the weight matrix, namely, from the row space, the null space, and the column space, and I analyzed the effect of the each space to Fréchet distances (FDs). Since the different spaces show different behaviors, choosing a subspace is not an insignificant decision. Instead of directly using the embedded distribution obtained from hidden layer's activations to calculate the FD, I proposed to use projection of embedded distribution onto the null space of the weight matrix among the three subspaces to avoid the fluctuations. My simulation results conducted at MNIST, CIFAR10, and CelebA datasets, show that, by projecting the embedded distributions onto the null spaces, possible parasitic effects coming from the randomness are being eliminated and reduces the number of needed simulations in MNIST dataset, in CIFAR10, and in CelebA dataset.

Low‐complexity linear precoding for low‐PAPR massive MU‐MIMO‐OFDM downlink systems

SummaryTo design new revolutionary wireless communication systems, orthogonal frequency‐division multiplexing (OFDM)‐based massive multiuser (MU) multiple‐input multiple‐output (MIMO) has been shown to be the most promising technology to significantly enhance the spectral, energy, and hardware efficiencies. However, massive MU‐MIMO‐OFDM transmitters exhibit signal with high peak‐to‐average power ratio (PAPR). Accordingly, the nonlinearity of the radio frequency (RF) power amplifier (PA), which causes the most severe hardware impairment, is expected to be a low‐cost and energy‐efficient component to enable cost‐ and energy‐efficient massive MU‐MIMO‐OFDM BS deployments, generating harmful in‐band distortion and out‐of‐band (OOB) emissions. In this paper, we develop a PAPR‐aware downlink transmission scheme in an OFDM‐based large‐scale MU‐MIMO. Linear precoding of data and peak‐canceling signals (PCSs) are employed to reduce the PAPRs of the transmitted signals by exploiting the excess degrees of freedom (DoFs) provided by equipping the base station (BS) by a large number of transmit antennas. Specifically, we design PCSs to be added to the frequency‐domain precoded data signals, with the goal of reducing the PAPRs of their time‐domain counterpart signals. Most importantly, the added PCSs have to lie in the null spaces of their associated MIMO channel matrices such that they do not cause any MU interference (MUI) and OOB radiation. In this regard, an efficient algorithm is developed, which is based on different data and PCSs precoders, and the corresponding achievable PAPR reduction and bit error rate (BER) performance are analyzed. Moreover, to optimize a tradeoff between performance and complexity, linear precoders based on matrix polynomials (M‐POLYs) and gradient‐iterative approaches are studied for both data and PCSs precoding. Simulation results reveal that these latter provide similar performance as the regularized zero‐forcing (RZF) and orthogonal projection null space (OPNS)‐based data and PCSs precoders, while they need much lower computational complexity. The substantial PAPR reduction provided by the proposed algorithm offers interesting insights for the design of energy‐efficient massive MU‐MIMO‐OFDM systems.

Study and analyze the eigenvalues and eigenvectors of a square matrix and study their applications through mathematical linear effects

In this paper we will investigate what properties are intrinsic to a matrix, or its associated linear application. As we will see, the fact that there are many bases in a vector space makes the expression of matrices or linear applications relative: it depends on which reference base we take. However, there are elements associated with this matrix, which do not depend on the reference base or bases that we choose, for example: a null spaces and a column spaces of a matrix, and their respective dimensions. The eigen values of matrix isa root of a character polynomial. Find a eigenvalue of matrix is equivalent to finding a rootof its polynomial. For matrices of size n ≥ 5, there does not generally existclosed expressions for the roots of the characteristic polynomial based on primary expressions (additions, subtractions, multiplications, divisions and roots). This result implies that the methods to find the valuesof a matrix must be iterativeOne way to calculate the eigen values would be to calculate the roots of thecharacteristic polynomial using a numerical method of calculating roots, like roots in Matlab. Roots in Python. But find the roots of a polynomial is usually a poorly conditioned problem. The conditioning of a problem has not been defined, but a wrong problemconditioned is a problem for which a small change in the data can induce an uncontrolled change in the results.