Triangular Matrix Research Articles

A note on a generalized Jordan form of an infinite upper triangular matrix

In this paper, several equivalent conditions for the existence of a generalized Jordan form for matrices of locally nilpotent linear operators acting on an infinite countable dimensional vector space are presented, and an example is given showing that these conditions are not always fulfilled, which means the existence of infinite upper triangular matrices that are not similar to any generalized infinite Jordan matrix. Besides, it is shown that these conditions are not applicable in the case of an arbitrary dimensional of a vector space.

A New Fixed Coefficient Diagonally Implicit Block Backward Differentiation Formula for Solving Stiff Initial Value Problems

Stiff initial value problems in ordinary differential equations occur when solution components evolve at varying rates, posing challenges for traditional computational methods. Specialized techniques are crucial for maintaining accuracy and stability during rapid transitions, emphasizing their significance in developing reliable numerical algorithms across scientific and engineering applications. This study aims to develop a new fixed coefficient 3-point diagonally implicit block backward differentiation formula for the numerical solution of first order stiff initial value problems. The method is constructed by integrating a triangular matrix into the coefficient matrix of an existing extended 3-point super class of block BDF for solving stiff initial value problems. The selection of a fixed coefficient within the interval accompanies this integration to ensure optimal stability. The method is found to order five. Stability analysis indicates that the method is consistent, zero-stable, and almost A-stable, validating its applicability to stiff initial value problems. Implementation of the method involves Newton’s iteration, and a code in the C programming language is devised to demonstrate its effectiveness. Comparative examination of numerical outcomes with the existing 3BBDF and 3ESBBDF methods highlights the proposed method's enhanced accuracy and reduced computation time.

Integer-valued polynomials on subsets of upper triangular matrix rings

S. Frisch, showed that the integer-valued polynomials on upper triangular matrix ring Int T n ( K ) ( T n ( D ) ) : = { f ∈ T n ( K ) [ x ] | f ( T n ( D ) ) ⊆ T n ( D ) } is a ring, where D is an integral domain with field of fractions K. Let R 1 ⊆ R 2 be commutative rings with identity. In this paper, we study the set Int T n ( R 2 ) ( Ω , T n ( R 1 ) ) : = { f ∈ T n ( R 2 ) [ x ] | f ( Ω ) ⊆ T n ( R 1 ) } for some subsets Ω ⊆ T n ( R 1 ) . We generalize Frisch’s result and show that Int T n ( R 2 ) ( T n ( R 1 ) ) : = Int T n ( R 2 ) ( T n ( R 1 ) , T n ( R 1 ) ) is a ring. We state a lower bound for the Krull dimension of the integer-valued polynomials on upper triangular matrix rings. Finally, we state the concept of Skolem closure of an ideal of the integer-valued polynomials on upper triangular matrix rings and as a consequence, we obtain a classification of maximal ideals of the integer-valued polynomials on upper triangular matrix rings.

Graded identities with involution for the algebra of upper triangular matrices

Let F be a field of characteristic zero and let m≥2 be an integer. In this paper, we prove that if a group grading on UTm(F) admits a graded involution then this grading is a coarsening of a Z⌊m2⌋-grading on UTm(F) and the graded involution is equivalent to the reflection or symplectic involution on UTm(F), this grading is called the finest grading on UTm(F). Furthermore, if m≤4 the algebra UTm(F) with the finest grading satisfies no non-trivial monomial identities. For the finest grading, a finite basis for the (Z⌊m2⌋,⁎)-identities is exhibited with the reflection and symplectic involutions and the asymptotic growth of the (Z⌊m2⌋,⁎)-codimensions is determined. As a consequence, we prove that for any G-grading on UTm(F) and any graded involution the (G,⁎)-exponent is m. Finally, we study the algebra UT3(F). For this algebra, there are, up to equivalence, two non-trivial gradings that admit a graded involution: the canonical Z-grading and the Z2-grading induced by (0,1,0). We determine a basis for the (Z2,⁎)-identities and we compute the codimension sequence for the (Z2,⁎)-graded identities for UT3(F).

Transposed Poisson structures on the Lie algebra of upper triangular matrices

We describe transposed Poisson structures on the upper triangular matrix Lie algebra T_{n}(F) , n>1 , over a field F of characteristic zero. We prove that, for n>2 , any such structure is either of Poisson type or the orthogonal sum of a fixed non-Poisson structure with a structure of Poisson type, and for n=2 , there is one more class of transposed Poisson structures on T_{n}(F) . We also show that, up to isomorphism, the full matrix Lie algebra M_{n}(F) admits only one non-trivial transposed Poisson structure, and it is of Poisson type.

A Triangular-Matrix-Based Spectral Encoding Method for Broadband Filtering and Reconstruction-Based Spectral Measurement.

Broadband filtering and reconstruction-based spectral measurement represent a hot technical route for miniaturized spectral measurement; the measurement encoding scheme has a great effect on the spectral reconstruction fidelity. The existing spectral encoding schemes are usually complex and hard to implement; thus, the applications are severely limited. Considering this, here, a simple spectral encoding method based on a triangular matrix is designed. The condition number of the proposed spectral encoding system is estimated and demonstrated to be relatively low theoretically; then, verification experiments are carried out, and the results show that the proposed encoding can work well under precise or unprecise encoding and measurement conditions; therefore, the proposed scheme is demonstrated to be an effective trade-off of the spectral encoding efficiency and implementation cost.

The new notion of Bohl dichotomy for non-autonomous difference equations and its relation to exponential dichotomy

In A. Czornik et al. [Spectra based on Bohl exponents and Bohl dichotomy for non-autonomous difference equations, J. Dynam. Differ. Equ. (2023)] the concept of Bohl dichotomy is introduced which is a notion of hyperbolicity for linear non-autonomous difference equations that is weaker than the classical concept of exponential dichotomy. In the class of systems with bounded invertible coefficient matrices which have bounded inverses, we study the relation between the set BD of systems with Bohl dichotomy and the set ED of systems with exponential dichotomy. It can be easily seen from the definition of Bohl dichotomy that ED ⊆ BD . Using a counterexample we show that the closure of ED is not contained in BD . The main result of this paper is the characterization int ⁡ BD = ED . The proof uses upper triangular normal forms of systems which are dynamically equivalent and utilizes a diagonal argument to choose subsequences of perturbations each of which is constructed with the Millionshikov Rotation Method. An Appendix describes the Millionshikov Rotation Method in the context of non-autonomous difference equations as a universal tool.

On normal complements

Although the theory of completion has a long history, there are not many results related to completing operator matrices to a normal operator. In this paper, we consider the problem of completion of the upper-triangular operator matrix [A?0B⁎] to normality, where A∈B(H) and B∈B(K) are called normal complements. We give different characterizations of normal complements and explore their joint spectral properties. We also investigate the similarity between A, B, and MC with respect to some essential properties such as regularity, Fredholmness, and others. Finally, we consider the self-duality property of Aluthge and Duggal transforms of hyponormal and semi-hyponormal operators.

A note on the image of polynomials on upper triangular matrix algebras

In the present paper we shall obtain the following result: Let n ≥ 2 and m ≥ 1 be integers. Let p ( x 1 , … , x m ) be a polynomial with zero constant term in non-commutative variables over an algebraically closed field K. Suppose that 1 ≤ r ≤ n − 1 , where r is the order of p. Then p ( T n ( K ) ) is a dense subset of T n ( K ) ( r − 1 ) (with respect to the Zariski topology). This is a solution of a problem presented by Gargate and de Mello and a supplement to a result obtained by Panja and Prasad in 2023.

Existence and uniqueness of solutions for Leontief's Input–Output Model, graph theory and sensitivity analysis

We provide a complete study of existence and uniqueness (uniqueness up to multiples in the case d = 0 ) of non-negative and non-trivial solutions x for the linear system ( I − A ) x = d with A ≥ 0 , d ≥ 0 (which, in particular, applies to Leontief's Input–Output Model). This study is done in terms of the block triangular form of the matrix A and is related to a directed graph associated to A . In particular, this study of existence and uniqueness up to multiples of solutions provides a framework that allows us to rigorously perform a sensitivity analysis of the normalized solutions of the linear system ( I − A ) x = d .

The generalized sectorial decompositions of semi-sectorial operators

This paper considers the generalized sectorial decompositions of semi-sectorial operators and quasi-sectorial operators on the complex Hilbert space. Under an invariant subspace condition, a necessary and sufficient condition for the existence of the generalized sectorial decomposition of a semi-sectorial operator is given. The generalized sectorial decomposition of a quasi-sectorial operator is given without any conditions. Some sufficient conditions for a block upper triangular operator matrix to be a semi-sectorial operator or a quasi-sectorial operator are given.

Emission Ghost Imaging: reconstruction with data augmentation.

Ghost Imaging enables 2D reconstruction of an object even though particles transmitted or emitted by the object of interest are detected with a single pixel detector without spatial resolution. This is possible because for the particular implementation of ghost imaging presented here, the incident beam is spatially modulated with a non-configurable attenuating mask whose orientation is varied (e.g. via transverse displacement or rotation) in the course of the ghost imaging experiment. Each orientation yields a distinct spatial pattern in the attenuated beam. In many cases, ghost imaging reconstructions can be dramatically improved by factoring the measurement matrix which consists of measured attenuated incident radiation for each of many orientations of the mask at each pixel to be reconstructed as the product of an orthonormal matrix and an upper triangular matrix provided that the number of orientations of the mask () is greater than or equal to the number of pixels () reconstructed. For the case, we present a data augmentation method that enables QR factorization of the measurement matrix. To suppress noise in the reconstruction, we determine the Moore-Penrose pseudoinverse of the measurement matrix with a truncated singular value decomposition approach. Since the resulting reconstruction is still noisy, we denoise it with the Adaptive Weights Smoothing method. In simulation experiments, our method outperforms a modification of an existing alternative orthogonalization method where rows of the measurement matrix are orthogonalized by the Gram-Schmidt method. We apply our ghost imaging methods to experimental X-ray fluorescence data acquired at Brookhaven National Laboratory.

A Simple Preprocessing Method Enhances Machine Learning Application to EEG Data for Differential Diagnosis of Autism

A new pre-processing approach of EEG data to detect topological EEG features has been applied to a continuous segment of artifact-free EEG data lasting 10 minutes in ASCII format derived from 50 ASD children and 50 children with other Neuro-Psychiatric Disorders (NPD), matched for age and male/female ratios. Each EEG is transformed in a triangular matrix of 171 values expressing all reciprocal Manhattan distances among the 19 electrodes of to the international 10-20 system. From this matrix, the minimum spanning tree (MST) is calculated. Electrode identification serial codes sorted according to the decreasing number of links in MST, and the number of links in MST are taken as input vectors for machine learning systems. Machine learning systems have been applied to build up a predictive model to distinguish between the two diagnostic classes (autism vs NPD) following a rigorous validation protocol. The best machine learning system (KNN algorithm) obtained a global accuracy of 93.2% (92.37 % sensitivity and 94.03 % specificity) in differentiating ASD subjects from NPD subjects. The results obtained in this study suggest that, thanks to the new pre-processing method introduced, there is the possibility to discriminate subjects with autism from subjects affected by other psychiatric disorders with a modest computational time reducing the information to 38 figures.

Dual modules, f-projective modules and Π-coherence over trivial ring extensions

Let R ⋉ M be a trivial extension of a ring R by an R-R-bimodule M. We first describe explicitly the structure of a dual module over R ⋉ M . Then we characterize Mittag-Leffler modules and f-projective modules over R × M. Finally, we investigate when R ⋉ M is a left Π-coherent ring. In addition, some applications are given in formal triangular matrix rings.

A note on products of idempotents in the ring of upper triangular infinite matrices

It is known that every singular matrix of a finite degree defined over a field can be written as a product of a finite number of idempotent matrices. Insipired by this result we investigate the same problem for N × N matrices. It turns out that in this case the analogous theorem does not hold.

Computing expected moments of the Rényi parking problem on the circle

A highly accurate and efficient method to compute the expected values of the count, sum, and squared norm of the sum of the centre vectors of a random maximal sized collection of non-overlapping unit diameter disks touching a fixed unit-diameter disk is presented. This extends earlier work on Renyi's parking problem [Magyar Tud. Akad. Mat. Kutato Int. Kozl. 3 (1–2), 1958, pp. 109–127]. Underlying the method is a splitting of the the problem conditional on the value of the first disk. This splitting is proven and then used to derive integral equations for the expectations. These equations take a lower block triangular form. They are solved using substitution and approximation of the integrals to very high accuracy using a polynomial approximation within the blocks.

Image Watermarking Using Discrete Wavelet Transform and Singular Value Decomposition for Enhanced Imperceptibility and Robustness

Digital multimedia elements such as text, image, audio, and video can be easily manipulated because of the rapid rise of multimedia technology, making data protection a prime concern. Hence, copyright protection, content authentication, and integrity verification are today’s new challenging issues. To address these issues, digital image watermarking techniques have been proposed by several researchers. Image watermarking can be conducted through several transformations, such as discrete wavelet transform (DWT), singular value decomposition (SVD), orthogonal matrix Q and upper triangular matrix R (QR) decomposition, and non-subsampled contourlet transform (NSCT). However, a single transformation cannot simultaneously satisfy all the design requirements of image watermarking, which makes a platform to design a hybrid invisible image watermarking technique in this work. The proposed work combines four-level (4L) DWT and two-level (2L) SVD. The Arnold map initially encrypts the watermark image, and 2L SVD is applied to it to extract the s components of the watermark image. A 4L DWT is applied to the host image to extract the LL sub-band, and then 2L SVD is applied to extract s components that are embedded into the host image to generate the watermarked image. The dynamic-sized watermark maintains a balanced visual impact and non-blind watermarking preserves the quality and integrity of the host image. We have evaluated the performance after applying several intentional and unintentional attacks and found high imperceptibility and improved robustness with enhanced security to the system than existing state-of-the-art methods.

Distributed fiber optic sensing for micro- and macro-crack quantification: an interfacial-fracture-energy-based model

Clarifying the strain transfer between the host material with a crack opening displacement (COD) and the optical fiber after interfacial debonding remains a critical challenge for crack quantification. The interfacial debonding induced a triangular form strain profile around the crack due to the residual shear stresses at the interface. This study investigated the interfacial shear stress-slip relationship between the bare fiber and its coating layer through a plate splitting test. Based on that, we proposed an interfacial-fracture-energy-based analytical model to convert distributed fiber optic strains before and after interfacial debonding to CODs of micro- and macro-cracks. The accuracy of the model under a single crack was validated through plate splitting tests and the experimental data reported in the literature. Furthermore, experimental results from a four-point beam test were utilized to investigate the feasibility of the model under multiple cracks. The model, assisted by the strain superposition strategy, accurately quantified the CODs of multiple cracks before interfacial debonding but failed when interfacial debonding occurred. In addition, we proposed a simplified method based on the linear relationship between the debonded lengths and the corresponding CODs, which facilitates evaluation of the CODs after debonding.

Integrated triangular fuzzy KE-GRA-TOPSIS method for dynamic ranking of products of customers’ fuzzy Kansei preferences

As society evolves, companies produce more homogeneous products, shifting customers’ needs from functionality to emotions. Therefore, how quickly customers select products that meet their Kansei preferences has become a key concern. However, customer Kansei preferences vary from person to person and are ambiguous and uncertain, posing a challenge. To address this problem, this paper proposes a TF-KE-GRA-TOPSIS method that integrates triangular fuzzy Kansei engineering (TF-KE) with Grey Relational Analysis (GRA) and Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS). Firstly, a Kansei evaluation system is constructed based on KE and fuzzy theory. A dynamic triangular fuzzy Kansei preference similarity decision matrix (TF-KPSDM) is defined to quantify customer satisfaction with fuzzy Kansei preferences. Secondly, dynamic objective weights are derived using Criteria Importance Though Intercrieria Correlation (CRITIC) and entropy, optimized through game theory to achieve superior combined weights. Thirdly, the GRA-TOPSIS method utilizes the TF-KPSDM and combined weights to rank products. Finally, taking the case of Kansei preference selection for electric bicycles, results indicate that the proposed method robustly avoids rank reversal and achieves greater accuracy than comparative models. This study can help companies dynamically recommend products to customers based on their Kansei preferences, increasing customer satisfaction and sales.

A Note on Singularity Categories and Triangular Matrix Algebras

A Note on Singularity Categories and Triangular Matrix Algebras