Inversion Of Matrices Research Articles

Periodicity-Aware Signal Denoising Using Capon-Optimized Ramanujan Filter Banks and Pruned Ramanujan Dictionaries

We propose a ‘periodicity-aware’ hybrid analysis-synthesis framework for denoising discrete-time periodic signals. Our method uses Ramanujan filter banks (RFB) for analysis and dictionaries for synthesis. The synthesis dictionary retains appropriate subspaces for signal reconstruction, by pruning the Ramanujan dictionary based on the outputs of the RFB. Unlike the other existing denoising methods, a unique advantage of the proposed method is that the denoised output is guaranteed to be composed of integer-periodic components with periods smaller than a pre-selected value. Our method works well even when the signal length is small, and has a high SNR gain over a wide range of input signal SNRs. Furthermore, we propose to adapt each filter in the analysis bank to the incoming data, by optimizing the filter coefficients through a multi-band Capon formulation. This helps in suppressing the spurious energy peaks generated from higher period filters in the analysis bank, further improving the denoising performance. Implementing multiband Capon filters requires inverses of several autocorrelation matrices. To reduce computations, a way to recursively compute these inverses based on Levinson's recursion is discussed. Next, we prove several multirate properties of Ramanujan subspace signals. An important property among these is that after decimation, a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$q$</tex-math></inline-formula> -th Ramanujan subspace signal still remains in the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$q$</tex-math></inline-formula> -th Ramanujan space, if and only if the decimation rate <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$M$</tex-math></inline-formula> is coprime to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$q$</tex-math></inline-formula> . This is helpful to further reduce computations required in the analysis part of the denoising framework by downsampling the filter outputs. Extensive Monte-Carlo simulations comparing different variants of the proposed method and several existing denoising methods are also provided.

Sharp partial order of core-nilpotent endomorphisms

The aim of this work is to offer a definition of the sharp partial order for the set of Core-Nilpotent endomorphisms. The main properties of this partial order are studied, generalizing the notion of sharp partial order of finite square matrices. In order to do so, the structure of the set of {g}-commuting inverses of a Core-Nilpotent endomorphism is described, and in particular, the structure of the set of {g}-commuting inverses of finite square matrices with entries in an arbitrary field is determined and an algorithm for its computation is provided.

A note on the Moore-Penrose inverse of block matrices

Motivated by the representation for the Moore-Penrose inverse of the block matrix over a *-regular ring presented in [R.E. Hartwig and P. Patr?cio, When does the Moore-Penrose inverse flip? Operators and Matrices, 6(1):181-192, 2012], we show that the formula of the Moore-Penrose inverse is the same as the expression given by [Nieves Castro-Gonz?lez, Jianlong Chen and Long Wang, Further results on generalized inverses in rings with involution, Elect. J. Linear Algebra, 30:118-134, 2015].

Pseudo Similarity of Neutrosophic Fuzzy matrices

In this paper, first we shall define Pseudo Similarity for Neutrosophic Fuzzy Matrices and prove that Pseudo Similarity relation on pair of Neutrosophic Fuzzy Matrices. Also, we derive some relation between Pseudo Similarity and Idempotent matrices. Finally, we give in varies inverse of Neutrosophic Fuzzy Matrices.

The group inverse of certain block complex matrices

We present new additive results for the group inverse of block complex matrices. As an application, the representations for the group inverse of a block complex matrix are given. These extend the main results of Ben?tez, Liu and Zhu (Linear Multilinear Algebra, 59(2011), 279-289).

An Algebraic Approach to n-Plithogenic Square Matrices For 𝟏𝟖 ≤ 𝒏 ≤ 𝟏𝟗

This paper is dedicated to study the algebraic properties and substructures that are related to the symbolic 18 and 19 plithogenic square matrices, where we present many different theorems and examples to clarify the computation of symbolic 18 and 19 plithogenic determinants, and finding 18-plithogenic and 19-plithogenic eigenvalues and eigenvectors. On the other hand, we provide algorithms for computing the inverses of symbolic 18-plithogenic and 19-plithogenic matrices with necessary and sufficient conditions for the orthogonality of those matrices.

A Note on Invertible Neutrosophic Square Matrices

The purpose of this article is to study the adjoint and inverse of neutrosophic matrices, where the inverse of a neutrosophic square matrix is defined and studied in terms of neutrosophic determinant and neutrosophic adjoint. It is shown by examples that, the converse part of the result “M is invertible if and only if detM ̸= 0” is not true, proved by Mohammad Abobala et al. in.2 Also some of the properties of neutrosophic adjoint are discussed.

A study of the equivalence of inference results in the contexts of true and misspecified multivariate general linear models

&lt;abstract&gt;&lt;p&gt;In practical applications of regression models, we may meet with the situation where a true model is misspecified in some other forms due to certain unforeseeable reasons, so that estimation and statistical inference results obtained under the true and misspecified regression models are not necessarily the same, and therefore, it is necessary to compare these results and to establish certain links between them for the purpose of reasonably explaining and utilizing the misspecified regression models. In this paper, we propose and investigate some comparison and equivalence analysis problems on estimations and predictions under true and misspecified multivariate general linear models. We first give the derivations and presentations of the best linear unbiased predictors (BLUPs) and the best linear unbiased estimators (BLUEs) of unknown parametric matrices under a true multivariate general linear model and its misspecified form. We then derive a variety of necessary and sufficient conditions for the BLUPs/BLUEs under the two competing models to be equal using a series of highly-selective formulas and facts associated with ranks, ranges and generalized inverses of matrices, as well as block matrix operations.&lt;/p&gt;&lt;/abstract&gt;

On the Discrete Implementation of the Filtered Inverse Method for Serial Robots

In this paper, the filtered inverse algorithm for dynamic inversion of matrices is revisited and additional analysis regarding its discrete implementation and related gain limitations are presented. The method is used to provide a solution to the inverse kinematics of a serial redundant robot manipulator with 7-DoF for both position and orientation control. The theoretical results obtained and the performance of the discrete method are validated on a relevant robotics simulation environment that also incorporates visual perception components, such as an eye-in-hand camera and artificial markers.

Characterizations of matrix equalities involving the sums and products of multiple matrices and their generalized inverse

&lt;abstract&gt;&lt;p&gt;It is common knowledge that matrix equalities involving ordinary algebraic operations of inverses or generalized inverses of given matrices can be constructed arbitrarily from theoretical and applied points of view because of the noncommutativity of the matrix algebra and singularity of given matrices. Two of such matrix equality examples are given by $ A_1B_1^{g_1}C_1 + A_2B_2^{g_2}C_2 + \cdots + A_kB_k^{g_k}C_{k} = D $ and $ A_1B_1^{g_1}A_2B_2^{g_2} \cdots A_kB_k^{g_k}A_{k+1} = A $, where $ A_1 $, $ A_2 $, $ \ldots $, $ A_{k+1} $, $ C_1 $, $ C_2 $, $ \ldots $, $ C_{k} $ and $ A $ and $ D $ are given, and $ B_1^{g_1} $, $ B_2^{g_2} $, $ \ldots $, $ B_k^{g_k} $ are generalized inverses of matrices $ B_1 $, $ B_2 $, $ \ldots $, $ B_k $. These two matrix equalities include many concrete cases for different choices of the generalized inverses, and they have been attractive research topics in the area of generalized inverse theory. As an ongoing investigation of this subject, the present author presents in this article several groups of new results and facts on constructing and characterizing the above matrix equalities for the mixed combinations of $ \{1\} $- and $ \{1, 2\} $-generalized inverses of matrices with $ k = 2, 3 $ by using some elementary methods, including a series of explicit rank equalities for block matrices.&lt;/p&gt;&lt;/abstract&gt;

An improved formula for the complete data fusion

Abstract. The complete data fusion is a method that combines independent measurements of atmospheric vertical profiles. Recently a new formula for the complete data fusion, which does not contain matrices that can be singular and overcomes the generalized inverse approximation used when singular matrices have to be inverted, has been proposed. We show that the new formula is a generalization of the original one and analyze the analytical relationship between the two formulas when generalized inverse matrices are used for the inversion of singular matrices. We extend the new formula to include interpolation and coincidence errors, which must be considered when the profiles to be fused are measured on different vertical grids and at different times and/or locations. Finally, we use a real measurement of the Infrared Atmospheric Sounding Interferometer (IASI) instrument to show the improved performances of the new formula with respect to the original one.

Semi-Supervised Non-Parametric Bayesian Modelling of Spatial Proteomics.

Understanding sub-cellular protein localisation is an essential component in the analysis of context specific protein function. Recent advances in quantitative mass-spectrometry (MS) have led to high resolution mapping of thousands of proteins to sub-cellular locations within the cell. Novel modelling considerations to capture the complex nature of these data are thus necessary. We approach analysis of spatial proteomics data in a non-parametric Bayesian framework, using K-component mixtures of Gaussian process regression models. The Gaussian process regression model accounts for correlation structure within a sub-cellular niche, with each mixture component capturing the distinct correlation structure observed within each niche. The availability of marker proteins (i.e. proteins with a priori known labelled locations) motivates a semi-supervised learning approach to inform the Gaussian process hyperparameters. We moreover provide an efficient Hamiltonian-within-Gibbs sampler for our model. Furthermore, we reduce the computational burden associated with inversion of covariance matrices by exploiting the structure in the covariance matrix. A tensor decomposition of our covariance matrices allows extended Trench and Durbin algorithms to be applied to reduce the computational complexity of inversion and hence accelerate computation. We provide detailed case-studies on Drosophila embryos and mouse pluripotent embryonic stem cells to illustrate the benefit of semi-supervised functional Bayesian modelling of the data.

Moore−Penrose inverse of the singular conditional matrices and its applications

Moore−Penrose inverse of the singular conditional matrices and its applications

Characterizations of the group invertibility of a matrix revisited

Abstract A square complex matrix A A is said to be group invertible if there exists a matrix X X such that A X A = A AXA=A , X A X = X XAX=X , and A X = X A AX=XA hold, and such a matrix X X is called the group inverse of A A . The group invertibility of a matrix is one of the fundamental concepts in the theory of generalized inverses, while group inverses of matrices have many essential applications in matrix theory and other disciplines. The purpose of this article is to reconsider the characterization problem of the group invertibility of a matrix, as well as the constructions of various algebraic equalities in relation to group invertible matrices. The coverage includes collecting and establishing a family of existing and new necessary and sufficient conditions for a matrix to be group invertible and giving many algebraic matrix equalities that involve Moore-Penrose inverses and group inverses of matrices through the skillful use of a series of highly selective formulas and facts about ranks, ranges, and generalized inverses of matrices, as well as block matrix operations.

Robust method for the identification of dynamical anisotropic flexible bearing parameters using multi-objective optimization and structural modification technique

In this paper, a deterministic multi-objective design optimization problem is addressed in order to identify the dynamical joint parameters such as mass, stiffness, damping and cross-coupling parameters. The proposed formulation uses directly the measured frequency transfer functions of the constrained system (with supports) and the mode of the unconstrained system (without supports) to predict the joint dynamic parameters. Since measurement noise often leads to erroneous identifications, the deterministic multi-objective optimization (DMOO) is used to eliminate the redistribution and amplification of noise due to the inversion of the frequency transfer function matrices. The proposed optimization design gives satisfactory results when using noise-free data as well as under realistic noise levels. A parametric study was also performed in order to assert the sensitivity of dynamical joints parameters against the level of noise. The results show also that the constrained non-dominated sorting genetic algorithm (C-NSGA-II) was capable to generate well-distributed reliable Pareto solutions.

Pivot invariance of multiconfiguration perturbation theory via frame vectors.

Multiconfiguration perturbation theory (MCPT) is a general framework for correcting the reference function of arbitrary structures. The variants of MCPT introduced so far differ in the specification of their zero-order Hamiltonian, i.e., the partitioning. A common characteristic of MCPT variants is that no numerical procedure is invoked when handling the overlap of the reference function and determinants spanning the configuration space. This comes at the price of pinpointing a principal term in the determinant expansion of the reference, rendering the PT results dependent on this choice. It is here shown that the pivot dependence of MCPT can be eliminated by using an overcomplete set of projected determinants in the space orthogonal and complementary to the reference. The projected determinants form a so-called frame, a generalization of the notion of basis, allowing for redundancy of the set. The simple structure of the frame overlap matrix facilitates overlap treatment in closed form, a feature shared by previous MCPT variants. In particular, the Moore-Penrose inverse of singular matrices appearing in frame-based MCPT can be constructed without the need for any pivoting algorithm or numerical zero threshold. Pilot numerical studies are performed for the singlet-triplet gap of biradicaloid systems, relying on geminal-based, incomplete model space reference function. Comparison with previous MCPT variants as well as illustration of pivot invariance is provided.

1MP and MP1 inverses and one-sided star orders in a ring with involution

The classes of 1MP-inverses and MP1-inverses are recently introduced classes of generalized inverses of complex matrices. Actually, they coincide with the classes of $$\{1,2,3\}$$ and $$\{1,2,4\}$$ inverses, respectively. We consider these inverses in the context of a ring with involution and prove that their most important characterizations and properties remain true. We show that the binary relations based on these inverses are in fact the well known left-star and right-star partial orders. We extend these relations to the ring case, connect them with the unified theory of partial order relations based on generalized inverses and provide several properties. Finally, we indicate how these results can be applied to bounded Hilbert space operators.

Graph-based algorithms for phase-type distributions

Phase-type distributions model the time until absorption in continuous or discrete-time Markov chains on a finite state space. The multivariate phase-type distributions have diverse and important applications by modeling rewards accumulated at visited states. However, even moderately sized state spaces make the traditional matrix-based equations computationally infeasible. State spaces of phase-type distributions are often large but sparse, with only a few transitions from a state. This sparseness makes a graph-based representation of the phase-type distribution more natural and efficient than the traditional matrix-based representation. In this paper, we develop graph-based algorithms for analyzing phase-type distributions. In addition to algorithms for state space construction, reward transformation, and moments calculation, we give algorithms for the marginal distribution functions of multivariate phase-type distributions and for the state probability vector of the underlying Markov chains of both time-homogeneous and time-inhomogeneous phase-type distributions. The algorithms are available as a numerically stable and memory-efficient open source software package written in C named ptdalgorithms. This library exposes all methods in the programming languages C and R. We compare the running time of ptdalgorithms to the fastest tools using a traditional matrix-based formulation. This comparison includes the computation of the probability distribution, which is usually computed by exponentiation of the sub-intensity or sub-transition matrix. We also compare time spent calculating the moments of (multivariate) phase-type distributions usually defined by inversion of the same matrices. The numerical results of our graph-based and traditional matrix-based methods are identical, and our graph-based algorithms are often orders of magnitudes faster. Finally, we demonstrate with a classic problem from population genetics how ptdalgorithms serves as a much faster, simpler, and completely general modeling alternative.

Group inverses of matrices of directed trees

A new class of directed trees is introduced. A formula for the group inverse of the matrices associated with any tree belonging to this class is obtained. This answers affirmatively, a conjecture of Catral et al., for this new class.

W-weighted GDMP inverse for rectangular matrices

In this article, we introduce two new generalized inverses for rectangular matrices called $W$-weighted generalized-Drazin--Moore--Penrose (GDMP) and $W$-weighted generalized-Drazin-reflexive (GDR) inverses. The first generalized inverse can be seen as a generalization of the recently introduced GDMP inverse for a square matrix to a rectangular matrix. The second class of generalized inverse contains the class of the first generalized inverse. We then exploit their various properties and establish that the proposed generalized inverses coincide with different well-known generalized inverses under certain assumptions. We also obtain a representation of $W$-weighted GDMP inverse employing EP-core nilpotent decomposition. We define the dual of $W$-weighted GDMP inverse and obtain analogue results. Further, we discuss additive properties, reverse- and forward-order laws for GD, $W$-weighted GD, GDMP, and $W$-weighted GDMP generalized inverses.