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Given a linear statistical model, statisticians have established many kinds of optimality criteria for estimating/predicting unknown parameters in the model using various mean square error methods. In this work, we revisit the well-known mean square error matrix (MSEM) for estimating/predicting unknown parameters in the context of general linear models from a completely new perspective. The coverage includes giving the definitions and analytical expressions of the minimum mean square error matrix predictor (MMSEMP) of a joint vector of all unknown parameters in the linear model by solving a quadratic matrix value optimization problem, and discussing a diverse of algebraic and statistical properties and additive decomposition equality of these predictors using some precise matrix analysis tools. [ABSTRACT FROM AUTHOR]

Algebraic expressions and equalities can be constructed arbitrarily in a given algebraic framework according to the operational rules provided, and thus it is a prominent and necessary task in mathematics and applications to construct, classify, and characterize various simple general algebraic expressions and equalities. As an update to this prominent topic in matrix algebra, this article reviews and improves upon the well-known block matrix methodology and matrix rank methodology in the construction and characterization of matrix equalities. We present a collection of fundamental and useful formulas for calculating the ranks of a wide range of block matrices and then derive from these rank formulas various valuable consequences. In particular, we present several groups of equivalent conditions in the characterizations of the Hermitian matrix, the skew-Hermitian matrix, the normal matrix, etc. [ABSTRACT FROM AUTHOR]

This article explores the mathematical and statistical performances and connections of the two well-known ordinary least-squares estimators (OLSEs) and best linear unbiased estimators (BLUEs) of unknown parameter matrices in the context of a multivariate general linear model (MGLM) for regression, both of which are defined under two different optimality criteria. Tian and Zhang [38] once collected a series of existing and novel identifying conditions for OLSEs to be BLUEs under general linear models: On connections among OLSEs and BLUEs of whole and partial parameters under a general linear model, Stat. Probabil. Lett., 112 (2016), 105–112. In this paper, we show how to extend this kind of results to multivariate general linear models. We shall give a direct algebraic procedure to derive explicit formulas for calculating the OLSEs and BLUEs of parameter spaces in a given MGLM, discuss the relationships between OLSEs and BLUEs of parameter matrices in the MGLM, establish many algebraic equalities related to the equivalence of OLSEs and BLUEs, and give various intrinsic statistical interpretations about the equivalence of OLSEs and BLUEs of parameter matrices in a given MGLM using some matrix analysis tools concerning ranks, ranges, and generalized inverses of matrices.

Many students who enroll in a first course in linear algebra major in STEM disciplines other than mathematics. Teachers who serve such students may find it difficult to provide authentic problems from these broader areas that ignite students' interest in linear algebra. In this paper, we highlight an interdisciplinary learning activity that engages students in using linear systems of equations to model the behavior of practical electric circuits. This exercise fits nicely into standard introductory linear algebra curricula and is designed to excite students majoring in engineering, physics, or applied mathematics. We also include references to a collection of open-access resources to support instructors who want to use this material in project-based, flipped-learning, inquiry-oriented, or independent-study environments. [ABSTRACT FROM AUTHOR]

In this paper, we define extensions of the classical eigenvalues of the matrix A ∈ Mm(C). These extensions are eigenvalues matrices for the block matrix A ∈ Mm(Mn), where Mm(Mn) is the set of all m×m block complex matrices with each block in Mn(C), they are.....Among other equalities and inequalities, we prove equalities which relate our new definitions with tr1(A) and tr2(A) as follows,... These relations are extensions of the classical relation tr(A) = Σmh=1λh(A), where A ∈Mm(C). Several new relations and properties of our new definitions are also given. [ABSTRACT FROM AUTHOR]

In 2012, we obtained Oppenheim's inequality for positive semidefinite matrices and its equality condition by the reproducing kernel method. In this paper, as a continuation, we give a reproducing kernel proof of the block matrix version of the Oppenheim-Schur's inequality and its equality condition in the positive definite case. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Jilin University (Science Edition) / Jilin Daxue Xuebao (Lixue Ban) is the property of Zhongguo Xue shu qi Kan (Guang Pan Ban) Dian zi Za zhi She and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Accurately aligning the same users on different flat social networks to merge user information and create more nuanced user profiles is critical. However, the current research in this area faces challenges related to low efficiency and inadequate alignment accuracy. To address these challenges, we introduce a cross-social network user alignment model based on multi-dimensional user features (MDUF). First, inspired by the principles of entity recognition and Hartley's association method, we employed a block matrix association algorithm to project the original dataset into different high-dimensional spaces. Second, we proposed a new inertia weight calculation method to improve the convergence speed from linear to nonlinear transformations. This method improves the performance of traditional particle swarm optimization algorithms. Finally, we utilize improved particle swarm optimization and residual connection techniques to optimize bidirectional long short-term memory networks. The experimental results show that our proposed model significantly outperforms traditional alignment models in terms of alignment efficiency and accuracy, which is highly practical and has the potential to inspire further research on social network user alignment.

In recent years, graph convolutional networks (GCNs) have gained significant attention for graph embedding learning. However, the efficacy of GCNs and their variants is often constrained by the implicit homophily assumption, which presumes nodes in close proximity to exhibit similar features. This assumption becomes particularly limiting in the context of complex heterogeneous graphs, as it restricts GCNs' ability to aggregate diverse node information. Addressing this, we propose a novel aggregation mechanism adept at discerning homophilic from heterophilic neighborhoods, thus achieving effective "classification aggregation". More specifically, this study combines block modeling with the aggregation process, allowing the GCNs to learn the aggregation rules across various classes of neighbors automatically. On this basis, a multi-view module is designed to extract semantic information from each view, and the final embedding is obtained by aggregating the information from all views through the attention module. Experiments indicated that the proposed method performs optimally across both heterogeneous and homogeneous graph datasets. [ABSTRACT FROM AUTHOR]

Let 0 < ρ ≤ 2 and w ρ (X) be the operator radius of a bounded linear Hilbert space operator X. In this paper we present characterizations of operators satisfying w ρ (X) ≤ 1. We also give an expression for the ρ -radii based on the numerical radius of a certain 2 × 2 block matrix. This enables us to investigate the properties of the operator radii w ρ (⋅). In particular, we obtain lower and upper bounds for the operator radii. In addition, we define a norm on Hilbert space operators, which generalizes the operator radii and prove some basic properties of this norm. Our results extend or improve some theorems due to Ando [4] , Bhatia–Jain [9] and Kittaneh [19]. [ABSTRACT FROM AUTHOR]

Algebraic expressions and equalities can be constructed arbitrarily in a given algebraic framework according to the operational rules provided, and thus it is a prominent and necessary task in mathematics and applications to construct, classify, and characterize various simple general algebraic expressions and equalities. As an update to this prominent topic in matrix algebra, this article reviews and improves upon the well-known block matrix methodology and matrix rank methodology in the construction and characterization of matrix equalities. We present a collection of fundamental and useful formulas for calculating the ranks of a wide range of block matrices and then derive from these rank formulas various valuable consequences. In particular, we present several groups of equivalent conditions in the characterizations of the Hermitian matrix, the skew-Hermitian matrix, the normal matrix, etc.

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.

In this paper, we are interested in generalized numerical radius inequalities for the off-diagonal part of a positive semidefinite block matrix. These inequalities produce a new set of inequalities for products and sums of matrices and some inequalities related to sectorial matrices. [ABSTRACT FROM AUTHOR]

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.

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 1 B 1 g 1 C 1 + A 2 B 2 g 2 C 2 + ⋯ + A k B k g k C k = D and A 1 B 1 g 1 A 2 B 2 g 2 ⋯ A k B k g k A k + 1 = A , where A 1 , A 2 , ... , A k + 1 , C 1 , C 2 , ... , C k and A and D are given, and B 1 g 1 , B 2 g 2 , ... , B k g k are generalized inverses of matrices B 1 , B 2 , ... , 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. [ABSTRACT FROM AUTHOR]

In this study, the problem of bundle adjustment was revisited, and a novel algorithm based on block matrix Cholesky decomposition was proposed to solve the thorny problem of self-calibration bundle adjustment. The innovation points are reflected in the following aspects: ① The proposed algorithm is not dependent on the Schur complement, and the calculation process is simple and clear; ② The complexities of time and space tend to O(n) in the context of world point number is far greater than that of images and cameras, so the calculation magnitude and memory consumption can be reduced significantly; ③ The proposed algorithm can carry out self-calibration bundle adjustment in single-camera, multi-camera, and variable-camera modes; ④ Some measures are employed to improve the optimization effects. Experimental tests showed that the proposed algorithm has the ability to achieve state-of-the-art performance in accuracy and robustness, and it has a strong adaptability as well, because the optimized results are accurate and robust even if the initial values have large deviations from the truth. This study could provide theoretical guidance and technical support for the image-based positioning and 3D reconstruction in the fields of photogrammetry, computer vision and robotics.

A square complex matrix AA is said to be group invertible if there exists a matrix XX such that AXA=AAXA=A, XAX=XXAX=X, and AX=XAAX=XA hold, and such a matrix XX is called the group inverse of AA. 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.

We first attempt to determine conditions on a linear relation T such that μT becomes a demicompact linear relation for each μ ∈ [0,1)(see Theorems 2.4 and 2.5). Second, we display some results on Fredholm and upper semi-Fredholm linear relations involving a demicompact one(see Theorems 3.1 and 3.2). Finally, we provide some results in which a block matrix of linear relations becomes a demicompact block matrix of linear relations (see Theorems 4.2 and 4.3).

One of the fundamental matrix equalities that involve multiple products of matrices and their generalized inverses is given by A 1 B 1 - A 2 B 2 - ... A k B k - A k + 1 = A , where A 1 , B 1 , A 2 , B 2 , ... , A k , B k , A k + 1 , and A are matrices of appropriate sizes, and B 1 - , B 2 - , ... , B k - are generalized inverses of matrices. Many matrix equality problems related to generalized inverses, including various reverse-order laws for generalized inverses of matrix products, can be reduced to the special situations of this general equality. Generally speaking, this equality does not necessarily hold, and therefore, it is a primary task to establish necessary and sufficient conditions for it to hold for some generalized inverses of matrices, or to always hold for all generalized inverses of matrices. The two basic cases of the equality problem for k = 1 , 2 and their special forms were well known and approached in the discipline of generalized inverses of matrices. As a further exploration on this kind of matrix, this note explores performances of the equality for k = 3 . We shall present an algebraic procedure to derive explicit necessary and sufficient conditions for the equality to always hold through the skilful use of rank equalities for some block matrices that are constructed from the given matrices, and then mentions the ideas and techniques of block matrix representations in the characterization of the above general matrix equality. [ABSTRACT FROM AUTHOR]

The study of constraining the eigenvalues of the sum of two symmetric matrices, say P + Q , in terms of the eigenvalues of P and Q , has a long tradition. It is closely related to estimating a lower bound on the minimum singular value of a matrix, which has been discussed by a great number of authors. To our knowledge, no study has yielded a positive lower bound on the minimum eigenvalue, λ min (P + Q) , when P + Q is symmetric positive definite with P and Q singular positive semi-definite. We derive two new lower bounds on λ min (P + Q) in terms of the minimum positive eigenvalues of P and Q. The bounds take into account geometric information by utilizing the Friedrichs angles between certain subspaces. The basic result is when P and Q are two non-zero singular positive semi-definite matrices such that P + Q is non-singular, then λ min (P + Q) ≥ (1 − cos ⁡ θ F) min ⁡ { λ min (P) , λ min (Q) } , where λ min represents the minimum positive eigenvalue of the matrix, and θ F is the Friedrichs angle between the range spaces of P and Q. Such estimates lead to new lower bounds on the minimum singular value of full rank 1 × 2 , 2 × 1 , and 2 × 2 block matrices. Some examples provided in this paper further highlight the simplicity of applying the results in comparison to some existing lower bounds. [ABSTRACT FROM AUTHOR]

This article introduces several new relations among related Hilbert space operators. In particular, we prove some Löewner partial orderings among T, |T|, RT, IT, |T|+|T∗| and many other related forms, as a new discussion in this field; where RT and IT are the real and imaginary parts of the operator T . Our approach will be based on proving the positivity of some new matrix operators, where several new forms for positive matrix operators will be presented as a key tool in obtaining the other ordering results. As an application, we present some results treating numerical radius inequalities in a way that extends some known results in this direction, in addition to some results about the singular values. [ABSTRACT FROM AUTHOR]

Let f (X 1 , X 2 , ... , X k) be a matrix function over the field of complex numbers, where X 1 , X 2 , ... , X k are a family of matrices with variable entries. The purpose of this paper is to propose and investigate the relationships between certain linear matrix functions that regularly appear in matrix theory and its applications. We shall derive a series of meaningful, necessary, and sufficient conditions for the collections of values of two given matrix functions to be equal through the cogent use of some highly selective formulas and facts regarding ranks, ranges, and generalized inverses of block matrix operations. As applications, we discuss some concrete topics concerning the algebraic connections between general solutions of a given linear matrix equation and its reduced equations. [ABSTRACT FROM AUTHOR]

Dynamic Time Warping (DTW) is a well-known similarity measure for time series. The standard dynamic programming approach to compute the DTW distance of two length-n time series, however, requires O (n 2) time, which is often too slow for real-world applications. Therefore, many heuristics have been proposed to speed up the DTW computation. These are often based on lower bounding techniques, approximating the DTW distance, or considering special input data such as binary or piecewise constant time series. In this paper, we present a first exact algorithm to compute the DTW distance of two run-length encoded time series whose running time only depends on the encoding lengths of the inputs. The worst-case running time is cubic in the encoding length. In experiments we show that our algorithm is indeed fast for time series with short encoding lengths. [ABSTRACT FROM AUTHOR]

This article has two interpenetrating motifs. One is an exposition of some major ideas and techniques behind the use of block matrices, and especially their positivity properties. This is done by focussing on one major problem: characterisation of operators whose numerical radius is bounded by one. So, the article could serve as an introduction to that topic as well. [ABSTRACT FROM AUTHOR]

Our aim is to present new expressions for the Drazin inverse of anti-triangular block matrices under some circumstances. Applying the established new formulae for anti-triangular block matrices, we derive explicit representations for the Drazin inverse of a 2×2 complex block matrix under corresponding assumptions. We extend several well known results in the literature in this way.

Consider the following problem: given a positive semidefinite (definite) matrix A ∈ Cmxm, and B ∈ Cnxm, such that ... = μ, find a matrix X satisfying subject to .... In 1996, Zheng presented the expressions for X formed by Löwner partial ordering when the problem is solvable, and pointed out that there may be no X satisfying these expressions. In this paper, we present some sufficient and necessary conditions for the existence of X and provide its explicit general expressions. [ABSTRACT FROM AUTHOR]

Several trace inequalities for positive semi-definite 2 × 2 block matrices are revisited. We present shorter proofs of these inequalities along with their refinements. We obtain bounds for the smallest and the largest eigenvalues of a positive semi-definite 2 × 2 block matrix involving the traces of its blocks. [ABSTRACT FROM AUTHOR]

Generalized inverses of matrices are extensions of ordinary inverses of nonsingular matrices to singular matrices, and people can construct various matrix expressions and equalities that involve matrices and their generalized inverses. This article considers a fundamental problem in theory of generalized inverses of matrices on characterizing range equalities composed of matrices and their generalized inverses. The coverage includes providing some general principles of dealing with this kind of problems, establishing a group of necessary and sufficient conditions for the equality range (D 1 - C 1 A 1 † B 1) = range (D 2 - C 2 A 2 † B 2) to hold for the Moore–Penrose generalized inverses A 1 † and A 2 † through the skillful use of the matrix rank method and the block matrix representation method, establishing a variety of range equalities with extrusion properties for multiple matrix products composed of two matrices and their conjugate transposes and Moore–Penrose generalized inverses, and deriving a group of novel necessary and sufficient conditions for the reverse-order law to hold via the matrix range equality method. [ABSTRACT FROM AUTHOR]

Our aim is to present new expressions for the Drazin inverse of anti-triangular block matrices under some circumstances. Applying the established new formulae for anti-triangular block matrices, we derive explicit representations for the Drazin inverse of a complex block matrix under corresponding assumptions. We extend several well known results in the literature in this way. [ABSTRACT FROM AUTHOR]

This paper studies the positive semi-definite 2×2 block matrix M=... ∈C2n×2n. A new sharp upper bound for |M| is provided under the condition of X being normal, for any unitarily invariant norm |·|. A special pattern among the eigenvalues of M when A+B=kI, for some k>0, is explored. [ABSTRACT FROM AUTHOR]

Given the linear matrix equation A X B = C , we partition it into the form A 1 X 11 B 1 + A 1 X 12 B 2 + A 2 X 21 B 1 + A 2 X 22 B 2 = C , and then pre- and post-multiply both sides of the equation by the four orthogonal projectors generated from the coefficient matrices A 1 , A 1 , B 1 , and B 2 to obtain four reduced linear matrix equations. In this situation, each of the four reduced equations involves just one of the four unknown submatrices X 11 , X 12 , X 21 , and X 22 , respectively. In this paper, we study the relationships between the general solution of A X B = C and the general solutions of the four reduced equations using some highly selective matrix analysis tools in relation to ranks, ranges, and generalized inverses of matrices. [ABSTRACT FROM AUTHOR]

The aim of this paper is to establish formulae for the Drazin inverse of anti-triangular block matrices under new assumptions in literature. Precisely, we consider the Drazin inverse of three kinds of anti-triangular block matrices. Applying these results, we present new expressions for the Drazin inverse of an arbitrary block matrix. In this way, we generalize a list of earlier results and illustrate it with three examples. [ABSTRACT FROM AUTHOR]

We first attempt to determine conditions on a linear relation T such that T becomes a demicompact linear relation for each 2 [0; 1) (see Theorems 2.4 and 2.5). Second, we display some results on Fredholm and upper semi-Fredholm linear relations involving a demicompact one (see Theorems 3.1 and 3.2). Finally, we provide some results in which a block matrix of linear relations becomes a demicompact block matrix of linear relations (see Theorems 4.2 and 4.3). [ABSTRACT FROM AUTHOR]

In this paper, we present some upper and lower bounds for operator space numerical radius of k×k block matrices, with entries inMn(X), when X is a numerical radius operator space. We also derive formulas for operator space numerical radius of some special bidiagonal block matrices. One of our main results states that if (X, (Wn)) is a numerical radius operator space and x1, . . ., xk ?Mn(X), then*** where ω = e 2πi/k is the k-th root of unity. [ABSTRACT FROM AUTHOR]

In this paper, the idea of partitioning is used to solve quaternion least squares problem, we divide the quaternion Bisymmetric matrix into four blocks and study the relationship between the block matrices. Applying this relation, the real representation of quaternion, and M-P inverse, we obtain the least squares Bisymmetric solution of quaternion matrix equation AXB=C and its compatable conditions. Finally, we verify the effectiveness of the method through numerical examples.

Block graphs are important baseline structures for a vast array of community detection and other network partitioning models. Singular graphs have many important uses in the physical sciences. A recent conjecture was posited that the nullity of a K 2 -free block graph cannot be larger than one. In this paper we prove that the conjecture is false by constructing a family of counterexamples using the Cauchy interlacing theorem for real symmetric matrices. In doing so, we prove the stronger statement that the nullity of K 2 -free block graphs is unbounded. Finally, the implications of this result for the computational network theory literature are discussed. [ABSTRACT FROM AUTHOR]

In this article, we construct new explicit formulas for the Drazin inverse of P + Q under some conditions, where P and Q are singular square matrices over skew fields. These formulas generalize some recent results in literature. By using these formulas, we establish new results for the Drazin inverse of 2 × 2 block matrix under some assumptions over skew fields. Finally, illustrative numerical examples over skew fields are presented to demonstrate results of this paper. [ABSTRACT FROM AUTHOR]

In this paper, a review of a development of the Drazin inverse for the sum of two matrices has been given. Since this topic is closely related to the problem of finding the Drazin inverse of a 2 × 2 block matrix, the paper also offers a survey of this subject. [ABSTRACT FROM AUTHOR]

Zhang and Xu recently obtained some new matrix norm inequalities in [1]. In this note, we provide alternative proofs and some applications for these results. [ABSTRACT FROM AUTHOR]

We present some unitarily invariant norm and Schatten p-norm inequalities relevant to accretive-dissipative matrices involving convex and concave functions. In particular, we show that if T = T i j 2 × 2 is an accretive-dissipative block matrix with T i j ∈ M n (C) , i , j = 1 , 2 , and f is an increasing convex function on [ 0 , ∞) with f (0) = 0 , then 2 f | T | 2 u ≤ ∥ f (2 | T 11 |) ∥ u + ∥ f (2 | T 22 |) ∥ u , for every unitarily invariant norm ∥ ⋅ ∥ u . We also extract several inequalities for accretive-dissipative n × n operator matrices. The obtained inequalities extend some known results. [ABSTRACT FROM AUTHOR]

Motivated by an application in high‐throughput genomics and metabolomics, we propose a novel and fully data‐driven approach for estimating large block structured sparse covariance matrices in the case where the number of variables is much larger than the number of samples without limiting ourselves to block diagonal matrices. Our approach consists in approximating such a covariance matrix by the sum of a low‐rank sparse matrix and a diagonal matrix. Our methodology also can deal with matrices for which the block structure appears only if the columns and rows are permuted according to an unknown permutation. Our technique is implemented in the R package BlockCov which is available from the Comprehensive R Archive Network (CRAN) and from GitHub. In order to illustrate the statistical and numerical performance of our package some numerical experiments are provided as well as a thorough comparison with alternative methods. Finally, our approach is applied to the use of 'multi‐omic' approaches for studying seed quality. [ABSTRACT FROM AUTHOR]

This article brings together miscellaneous formulas and facts on matrix expressions that are composed by idempotent matrices in one place with cogent introduction and references for further study. The author will present the basic mathematical ideas and methodologies of the matrix analytic theory in a readable, up-to-date, and comprehensive manner, including constructions of various algebraic matrix identities composed by the conventional operations of idempotent matrices, and uses of the block matrix method in the derivation of closed-form formulas for calculating the ranks of matrix expressions that are composed by idempotent matrices. The author also determines the maximum and minimum ranks of some matrix pencils composed by the products of matrices and their generalized inverses and uses the ranks to characterize algebraic performance of the matrix pencils.

Let f(X1,X2,…,Xk) be a matrix function over the field of complex numbers, where X1,X2,…,Xk are a family of matrices with variable entries. The purpose of this paper is to propose and investigate the relationships between certain linear matrix functions that regularly appear in matrix theory and its applications. We shall derive a series of meaningful, necessary, and sufficient conditions for the collections of values of two given matrix functions to be equal through the cogent use of some highly selective formulas and facts regarding ranks, ranges, and generalized inverses of block matrix operations. As applications, we discuss some concrete topics concerning the algebraic connections between general solutions of a given linear matrix equation and its reduced equations.

Copyright of Telecommunication Engineering is the property of Telecommunication Engineering and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)