Multilinear Algebra Research Articles

Advancements in the generalization of m-isometries: introducing the q-partial-(m, A)-isometry class

In this paper, we extend the concepts of m-partial isometry of order q and ( m , A ) -isometry, building on the contributions of Aouichaoui MA. [A note on partial-A-isometries and some applications. Quaest Math. 2024; 47(3):515–535], Bermúdez et al. [ ( m , A ) -isometries on Hilbert spaces. Linear Algebra Appl. 2018;540:95–111], Sadi A, Mahmoudi F. [ ( m , A ) -partial isometries in semi-Hilbertian spaces. Linear Multilinear Algebra. doi: 10.1080/03081087.2022.2068493], Sid Ahmed OAM, Saddi A. [A-m-Isometric operators in semi-Hilbertian spaces. Linear Algebra Appl. 2012;436(10):3930–3942], Sid Ahmed OAM. [Generalization of m-partial isometries on a Hilbert space. Int J Pure Appl Math. 2015;104(4):599–619]. Specifically, we introduce and define the class of q-partial- ( m , A ) -isometries, and provide a detailed analysis of their matrix representations. Additionally, we explore their spectral properties.

High-resolution direction-finding algorithm using Qth−order cumulants tensor: Q−TMUSIC

To resolve more sources than sensors, the 2q−MUSIC algorithm has been developed recently by rearranging the 2qth−order circular cumulants tensor into a suitable size matrix. However, in 2q−MUSIC, the angular resolution and estimation accuracy have not attained their optimum values due to the loss of differential information by matricizing the cumulants tensor. This paper proposes a subspace-based MUSIC-like algorithm, namely Qth−order cumulants-based tensor MUSIC algorithm for the direction of arrival (DOA) estimation of statistically independent non-Gaussian sources by exploiting the Qth−order cumulants tensors, Q∈{3,4,5,⋯}, directly using multilinear algebra tools. The proposed algorithm offers an improved DOA estimation performance in terms of angular resolution and estimation accuracy compared to their equivalent matrix-based Q−MUSIC. Moreover, the computational complexity of the proposed Q-TMUSIC is lesser than that of their equivalent matrix-based Q−MUSIC for Q>4. However, the proposed Q−TMUSIC can identify atmost (M−1) sources using an array of M−sensors than that of (M⌊Q2⌋−1) with their equivalent matrix based Q−MUSIC.

Entries of the bottleneck matrices of maximal outerplanar graphs

In this paper, we consider the entries of the bottleneck matrices of maximal outerplanar graphs. We recall patterns from [Kirkland SJ, Neumann M, Shader BL. Characteristic vertices of weighted trees via Perron values. Linear Multilinear Algebra. 1996;40:311–325.] involving the entries of bottleneck matrices for trees. We first establish similar patterns involving the diagonal entries of bottleneck matrices for maximal outerplanar graphs. We then investigate patterns involving the off-diagonal entries of bottleneck matrices for maximal outerplanar graphs and establish similar results to those found in [Kirkland SJ, Neumann M, Shader BL. Characteristic vertices of weighted trees via Perron values. Linear Multilinear Algebra. 1996;40:311–325.] for trees. We close with examples illustrating our results.

An improvement and a generalization of Rotfel'd type inequalities for sectorial matrices

By using equivalence conditions for sectorial matrices obtained by Alakhrass and Sababheh in Linear Algebra Appl. (2020;586), we improve a Rotfel'd type inequality for sectorial matrices derived by Zhang in Linear Multilinear Algebra (2015;63) and generalize a recent result of Mao and Ji in Linear Algebra Appl. (2024;691).

On affine spaces of alternating matrices with constant rank

Let F be a field, and n ≥ r > 0 be integers, with r even. Denote by A n ⁡ ( F ) the space of all n-by-n alternating matrices with entries in F . We consider the problem of determining the greatest possible dimension for an affine subspace of A n ⁡ ( F ) in which every matrix has rank equal to r (or rank at least r). Recently Rubei [Affine subspaces of skewsymmetric matrices with constant rank. Linear Multilinear Algebra. 2023. in press. doi: 10.1080/03081087.2023.2198759.] has solved this problem over the field of real numbers. We extend her result to all fields with large enough cardinality. Provided that n ≥ r + 3 and | F | ≥ min ( r − 1 , r 2 + 2 ) , we also determine the affine subspaces of rank r matrices in A n ⁡ ( F ) that have the greatest possible dimension, and we point to difficulties for the corresponding problem in the case n ≤ r + 2 .

Optimality of generalized Choi maps in M 3

A family of linear positive maps in the algebra of 3×3 complex matrices proposed recently by Bera et al 2024 Linear and Multilinear Algebra 1–16) is further analyzed. It provides a generalization of a seminal Choi nondecomposable extremal map in M 3. We investigate when generalized Choi maps are optimal, i.e. cannot be represented as a sum of positive and completely positive maps. This property is weaker than extremality, however, it turns out that it plays a key role in detecting quantum entanglement.

A note on Zhu’s construction of Hopf brace from Radford biproducts

In this paper, a new construction of Hopf brace from Radford biproducts is investigated, which is essentially different from the construction in [Linear Multilinear Algebra 70 (2022) 3171–3188].

Inertia of partial transpose of positive semidefinite matrices

We show that the partial transpose of 9×9 positive semidefinite matrices do not have inertia (4,1,4) and (3,2,4) . It solves an open problem in ‘LINEAR AND MULTILINEAR ALGEBRA. Changchun Feng et al, 2022’. We apply our results to construct some inertia, as well as present the list of all possible inertia of partial transpose of 12×12 positive semidefinite matrices.

Linear to multi-linear algebra and systems using tensors

In the past few decades, multi-linear algebra also known as tensor algebra has been adapted and employed as a tool for various engineering applications. Recent developments in tensor algebra have indicated that several well-known concepts from linear algebra can be extended to a multi-linear setting with the help of a special form of tensor contracted product, known as the Einstein product. Thus, the tensor contracted product and its properties can be harnessed to define the notions of multi-linear system theory where the input, output signals, and the system are inherently multi-domain or multi-modal. This study provides an overview of tensor algebra tools which can be seen as an extension of linear algebra, at the same time highlighting the differences and advantages that the multi-linear setting brings forth. In particular, the notions of tensor inversion, tensor singular value, and tensor eigenvalue decomposition using the Einstein product are explained. In addition, this study also introduces the notion of contracted convolution for both discrete and continuous multi-linear system tensors. Tensor network representation of various tensor operations is also presented. In addition, application of tensor tools in developing transceiver schemes for multi-domain communication systems, with an example of MIMO CDMA system, is presented. This study provides a foundation for professionals whose research involves multi-domain or multi-modal signals and systems.

Jointly $ A $-hyponormal $ m $-tuple of commuting operators and related results

<p>In this paper, we aim to investigate the class of jointly hyponormal operators related to a positive operator $ A $ on a complex Hilbert space $ \mathcal{X} $, which is called jointly $ A $-hyponormal. This notion was first introduced by Guesba et al. in [Linear and Multilinear Algebra, 69(15), 2888–2907] for $ m $-tuples of operators that admit adjoint operators with respect to $ A $. Mainly, we prove that if $ \mathbf{B} = (B_1, \cdots, B_m) $ is a jointly $ A $-hyponormal $ m $-tuple of commuting operators, then $ \mathbf{B} $ is jointly $ A $-normaloid. This result allows us to establish, for a particular case when $ A $ is the identity operator, a sharp bound for the distance between two jointly hyponormal $ m $-tuples of operators, expressed in terms of the difference between their Taylor spectra. We also aim to introduce and investigate the class of spherically $ A $-$ p $-hyponormal operators with $ 0 < p < 1 $. Additionally, we study the tensor product of specific classes of multivariable operators in semi-Hilbert spaces.</p>

ℤ-gradings on the Grassmann algebra over infinite fields: Graded identities and central polynomials

Let E be the infinite-dimensional Grassmann algebra over an infinite field F of characteristic different from 2. The main purpose of this paper is to describe the [Formula: see text]-ideal of the graded polynomial identities and the [Formula: see text]-space of the central polynomials of E equipped with its [Formula: see text] and 3-induced [Formula: see text]-gradings. Therefore, we generalize the results of [A. Brandão, C. Fidelis and A. Guimarães, [Formula: see text]-gradings of full support on the Grassmann algebra, J. Algebra 601 (2022) 332–353; C. Fidelis, A. Guimarães and P. Koshlukov, A note on [Formula: see text]-gradings on the Grassmann algebra and elementary number theory, Linear Multilinear Algebra 71(7) (2023) 1244–1264] in the PI theory context.

Multilinear algebra methods for higher-dimensional graphs

In this paper, we will explore the use of multilinear algebra-based methods for higher dimensional graphs. Multi-view clustering (MVC) has gained popularity over the single-view clustering due to its ability to provide more comprehensive insights into the data. However, this approach also presents challenges, particularly in combining and utilizing multiple views or features effectively. Most of recent work in this field focuses mainly on tensor representation instead of treating the data as simple matrices. Accordingly, we will examine and compare these approaches, particularly in two categories, namely graph-based clustering and subspace-based clustering. We will report on experiments conducted using benchmark datasets to evaluate the performance of the main clustering methods.

Tensor-Based Approaches for Nonlinear and Multilinear Systems Modeling and Identification

Nonlinear (NL) and multilinear (ML) systems play a fundamental role in engineering and science. Over the last two decades, active research has been carried out on exploiting the intrinsically multilinear structure of input–output signals and/or models in order to develop more efficient identification algorithms. This has been achieved using the notion of tensors, which are the central objects in multilinear algebra, giving rise to tensor-based approaches. The aim of this paper is to review such approaches for modeling and identifying NL and ML systems using input–output data, with a reminder of the tensor operations and decompositions needed to render the presentation as self-contained as possible. In the case of NL systems, two families of models are considered: the Volterra models and block-oriented ones. Volterra models, frequently used in numerous fields of application, have the drawback to be characterized by a huge number of coefficients contained in the so-called Volterra kernels, making their identification difficult. In order to reduce this parametric complexity, we show how Volterra systems can be represented by expanding high-order kernels using the parallel factor (PARAFAC) decomposition or generalized orthogonal basis (GOB) functions, which leads to the so-called Volterra–PARAFAC, and Volterra–GOB models, respectively. The extended Kalman filter (EKF) is presented to estimate the parameters of a Volterra–PARAFAC model. Another approach to reduce the parametric complexity consists in using block-oriented models such as those of Wiener, Hammerstein and Wiener–Hammerstein. With the purpose of estimating the parameters of such models, we show how the Volterra kernels associated with these models can be written under the form of structured tensor decompositions. In the last part of the paper, the notion of tensor systems is introduced using the Einstein product of tensors. Discrete-time memoryless tensor-input tensor-output (TITO) systems are defined by means of a relation between an Nth-order tensor of input signals and a Pth-order tensor of output signals via a (P+N)th-order transfer tensor. Such systems generalize the standard memoryless multi-input multi-output (MIMO) system to the case where input and output data define tensors of order higher than two. The case of a TISO system is then considered assuming the system transfer is a rank-one Nth-order tensor viewed as a global multilinear impulse response (IR) whose parameters are estimated using the weighted least-squares (WLS) method. A closed-form solution is proposed for estimating each individual IR associated with each mode-n subsystem.

Biderivations of complete Lie algebras

The authors of this article intend to present some results obtained in the study of biderivations of complete Lie algebras. Firstly they present a matricial approach to do this, which was a useful and explanatory tool not only in the study of biderivations but also in the synthesis of these results. Then they study all biderivations of a Lie algebra [Formula: see text] with [Formula: see text] and [Formula: see text], called complete. Moreover, as an application of the previous result, they describe all biderivations of a semisimple Lie algebra (that are complete), extending a result obtained by Tang in [X. Tang, Biderivations of finite-dimensional complex simple Lie algebras, Linear Multilinear Algebra 66(2) (2018) 250–259] that describes all biderivations of a complex simple Lie algebra. And thirdly, results on symmetric and skew-symmetric biderivations are also presented.

Symmetric closure in modules and rings

We introduce both for modules and rings classes of elements that are strongly connected to commutativity classes as defined in Abdi, M., Leroy, A. G. (2021). Graphs of commutatively closed sets. Linear Multilinear Algebra. 70(21):6965–6977 and Alghazzawi, D., Leroy, A. G. Commutatively closed sets in rings. Commun. Algebra. 47(4):1629–1641. We define a graph structure on the classes leading to a notion of distance for elements of a class. Examples are presented all along the paper, showing the interest of the notions.

Attention U-Net Based on Bi-ConvLSTM and Its Optimization for Smart Healthcare

As an important part of cyber–physical–social intelligence, artificial intelligence (AI)-driven smart healthcare is committed to promoting the application of human–machine hybrid augmented intelligence in the medical field, including AI-assisted medical image analysis and lesion recognition. Among them, deep learning models represented by fully convolutional networks (FCNs) have achieved excellent performance in medical image segmentation. However, limited by the complex structure of segmentation networks and the inherently redundant characteristics of convolutional operation, the scale of these models is extremely large. To further promote the application of machine intelligence in the field of medical image analysis, we propose an attention U-Net based on Bi-ConvLSTM (AUBC-Net) for accurate segmentation of medical images in this article. Different from classical U-Net, the proposed model deals with the potential association between decoding features and encoding features by bidirectional convolution LSTM. Furthermore, for the inherent redundancy characteristics of FCNs, we propose a lightweight feature generation strategy and optimize the calculation process of Bi-ConvLSTM based on tensor multilinear algebra, which can greatly reduce the number of network parameters. In addition, we have conducted the image segmentation experiments on two benchmark medical datasets, and the experimental results demonstrate that the proposed model can not only achieve better performance than existing methods, but also effectively compress network parameters while ensuring performance, which greatly facilitates AI-driven smart medical applications.

Unsupervised dimensionality reduction of medical hyperspectral imagery in tensor space

Background and objectivesCompared with traditional RGB images, medical hyperspectral imagery (HSI) has numerous continuous narrow spectral bands, which can provide rich information for cancer diagnosis. However, the abundant spectral bands also contain a large amount of redundancy information and increase computational complexity. Thus, dimensionality reduction (DR) is essential in HSI analysis. All vector-based DR methods ignore the cubic nature of HSI resulting from vectorization. To overcome the disadvantage of vector-based DR methods, tensor-based techniques have been developed by employing multi-linear algebra. MethodsTo fully exploit the structure features of medical HSI and enhance computational efficiency, a novel method called unsupervised dimensionality reduction via tensor-based low-rank collaborative graph embedding (TLCGE) is proposed. TLCGE introduces entropy rate superpixel (ERS) segmentation algorithm to generate superpixels. Then, a low-rank collaborative graph weight matrix is constructed on each superpixel, greatly improving the efficiency and robustness of the proposed method. After that, TLCGE reduces dimensions in tensor space to well preserve intrinsic structure of HSI. ResultsThe proposed TLCGE is tested on cholangiocarcinoma microscopic hyperspectral data sets. To further demonstrate the effectiveness of the proposed algorithm, other machine learning DR methods are used for comparison. Experimental results on cholangiocarcinoma microscopic hyperspectral data sets validate the effectiveness of the proposed TLCGE. ConclusionsThe proposed TLCGE is a tensor-based DR method, which can maintain the intrinsic 3-D data structure of medical HSI. By imposing the low-rank and sparse constraints on the objective function, the proposed TLCGE can fully explore the local and global structures within each superpixel. The computational efficiency of the proposed TLCGE is better than other tensor-based DR methods, which can be used as a preprocessing step in real medical HSI classification or segmentation.

On structural and spectral properties of reduced power graph of finite groups

The reduced power graph of a finite group [Formula: see text], denoted by [Formula: see text], is the graph whose vertices are the elements of the group [Formula: see text] and two vertices [Formula: see text] are adjacent if and only if [Formula: see text] or [Formula: see text]. In this paper, we first describe the structure of the reduced power graph of the finite cyclic group [Formula: see text]. Consequently, we provide a short and alternative proof of one of the results published in (R. Rajkumar and T. Anitha, Laplacian spectrum of reduced power graph of certain finite groups, Linear Multilinear Algebra (2019) 1–18). We characterize the values of [Formula: see text] for which [Formula: see text] is a line graph. We then deduce the signless Laplacian spectrum of [Formula: see text] using its structure. We provide lower and upper bounds on the signless Laplacian spectral radius of [Formula: see text]. Finally, we conclude the paper by determining the signless Laplacian spectrum of [Formula: see text], where [Formula: see text] denotes the dihedral group of order [Formula: see text].

Decompositions of endomorphisms into a sum of roots of the unity and nilpotent endomorphisms of fixed nilpotence

For n≥2 and fixed k≥1, we study when an endomorphism f of Fn, where F is an arbitrary field, can be decomposed as t+m where t is a root of the unity endomorphism and m is a nilpotent endomorphism with mk=0. For fields of prime characteristic, we show that this decomposition holds as soon as the characteristic polynomial of f is algebraic over its base field and the rank of f is at least nk, and we present several examples that show that the decomposition does not hold in general. Furthermore, we completely solve this decomposition problem for k=2 and nilpotent endomorphisms over arbitrary fields (even over division rings). This somewhat continues our recent publications in Linear Multilinear Algebra (2022) and Int. J. Algebra Comput. (2022) as well as it strengthens results due to Cǎlugǎreanu-Lam in J. Algebra Appl. (2016).

On the circular numerical range of 5-by-5 partial isometries

We prove, in some cases in terms of kippenhahn curve, that if 5-by-5 partial isometry whose numerical range is a circular disc then its center must be the origin. This gives a partial affirmative answer to Conjecture 5.1. of [H. l. Gau et al., Linear and Multilinear Algebra, 64 (1) 2016, 14–35.], for the five-dimensional case.