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Journal articles on the topic 'Matrices triangulares'

Author: Grafiati

Published: 5 June 2025

Last updated: 24 June 2025

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1

Rodríguez González, Ihosvany, and Anié Bermudez Peña. "Resolución paralela de sistemas triangulares." Innovación y Software 1, no. 2 (2020): 27–39. http://dx.doi.org/10.48168/innosoft.s2.a25.

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La resolución de sistemas triangulares es un núcleo computacional ampliamente utilizado en diversas aplicaciones científicas. Esta investigación realiza la implementación y comparación de varios algoritmos paralelos frente a un algoritmo secuencial eficiente para la resolución de sistemas triangulares. Los algoritmos se distinguen por la forma de particionado de la matriz y la asignación a los procesadores. Se realiza el análisis del comportamiento de los algoritmos en la solución de sistemas de ecuaciones lineales triangulares superiores en un clúster de computadoras. Para ello se tienen en cuenta las métricas de tiempo aritmético, tiempo de comunicaciones, aceleración y eficiencia máxima. Se realizaron experimentos para cada algoritmo con distintos tamaños de matrices sobre varios procesadores. El algoritmo con mejores resultados fue el que divide por bloques las filas de la matriz y aplica una distribución cíclica en el cluster.

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2

José, Figueroa, Villarroel Felicia, Ramı́rez Henry, and Rosas Tobı́as. "Matriz de adyacencia de Ramsey del grafo K_R(G,H) con componentes h-buenas y las relaciones geométricas entre lados y vértices de los grafos G, H y K_R(G,H)." Divulgaciones Matemáticas 22, no. 2 (2022): 34——47. https://doi.org/10.5281/zenodo.7487440.

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Sean \(G\) y \(H\) dos grafos simples, finitos, y no vacíos. El número de Ramsey \(R(G,H)\), se define como el menor entero positivo \(n\), tal que hay un grafo \(F\) de orden \(n\) que contiene un subgrafo \(G'\) copia monocromática isomorfa a \(G\), o el complemento de \(F\) (denotado por \(\overline{F}\)) contiene un subgrafo \(H'\) copia monocromática isomorfa a \(H\). Se dice que el grafo completo \(K_{n}\) contiene componentes \(h\)-buena, si para toda secuencia \(s_{i}\), con \(i={1,\cdots ,m+1}\), donde \(m\) es la talla de cada secuencia que colorean los lados del grafo completo \(K_{n}=F\cup\overline{F}\), tal que se pueda extraer de \(F\), al menos una copia monocromática \(G'\) isomorfa a\(G\), o \(\overline{F}\) contenga al menos una copia monocromática \(H'\) isomorfa a \(H\). En este manuscrito se presentan dos resultados principales, a saber: 1) Se determinan los lados incidentes de cada vértices \(v_{1},\ldots, v_{n}\) del grafo \(G\) de orden \(n\), a través de su matriz de adyacencia \(A(G)\), obteniendo la fórmula \(Traz(M)=\sum_{i=1}^{n} d(v_{i})=2|E(G)|\), donde \(M=\left(A(G)\right)^{2}\) es una matriz de orden \(n\times n\), \(Traz(M)\) es la traza de la matriz \(M\), y \(d(v_{i})\) es el grado del vértice \(v_{i}\) para \(i=1,\ldots,n\). 2) Se determina la matriz de adyacencia Ramsey del menor grafo completo \(K_{R(G,H)}\) con componentes \(h\)-buena. Se determina a través de los elementos \(m_{ij}\) de \(M\), las relaciones existentes entre los lados y los vértices de los grafos \(G\) y \(H\), con respecto a \(K_{R(G,H)}\) y se obtuvieron las siguientes propiedades: \(\displaystyle{\sum_{i>j}m_{ij}=\sum_{i<j}m_{ij}=m_{ij}|E(K_{R(G,H)})|=k|E(K_{R(G,H)})|}\) con \(k=m_{ij}\in M\) Existen \(r,s\in \mathbb{Z}^{+}\), dependientes de \(E(G), E(H)\) y \(E(K_{R(G,H)})\), tal que \(\dfrac{E(K_{R(G,H)})}{r}=\dfrac{E(G)}{s}\). Existen \(p,q\in \mathbb{Z}^{+}\), dependientes de \(V(G), V(H)\) y \(V(K_{R(G,H)})\), tal que, \(\dfrac{V(K_{R(G,H)})}{p}=\dfrac{V(H)}{q}\). \(\displaystyle{Traz(M)=\sum_{i=1}^{n} d(v_{i})=2|E(K_{n})|}\).

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3

Aryani, Fitri, Medyantiwi Rahmawita, Megawati, and Sarbaini. "The Formula of the Trace of Triangle n×n Matrix to the Power of Positive Integer." European Journal of Mathematics and Statistics 4, no. 4 (2023): 24–37. http://dx.doi.org/10.24018/ejmath.2023.4.4.248.

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This study determined the general form of the trace of the triangular matrices n × n with the power of positive integer. Before obtaining the general form of the trace of triangular matrices (upper triangle and lower triangle) n × n with the power positive integer, first obtain the general form of the triangular matrices n × n with power positive integer. Obtaining the general form of the triangular matrices n × n with the power positive integer is carried out by determining of the triangular matrices from power two to power eight. It is further suspected that the general form of a triangular matrices n × n with the power of a positive integer and prove it using mathematical induction. Finally, a triangular matrices trace n × n with the power of a positive integer is obtained with direct proof based on the general form of the matrices has been obtained. Given the application trace of the triangle matrices n × n with power positive integer by an example.

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4

Hoffmann, W., and J. J. B. De Swart. "Approximating Runge-Kutta matrices by triangular matrices." BIT Numerical Mathematics 37, no. 2 (1997): 346–54. http://dx.doi.org/10.1007/bf02510217.

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5

Ma, Chao, Yali Ren, Zheng Li, and Jin Zhong. "Matrices whose powers are eventually triangular." Filomat 37, no. 26 (2023): 8867–85. http://dx.doi.org/10.2298/fil2326867m.

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Square matrices whose powers eventually have some special properties are of both theoretical significance and application value. This paper investigates those complex matrices whose powers are eventually triangular. We completely characterize the eventually triangular complex matrices of order not greater than 4, and extend the results to the nonnegative case. Eventually triangular matrices of order n are also discussed.

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6

DeFranza, J., and D. J. Fleming. "Positioning matrices with respect to the boundary of the maximal group." Bulletin of the Australian Mathematical Society 33, no. 1 (1986): 113–22. http://dx.doi.org/10.1017/s0004972700002938.

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Let Δ denote the Banach algebra of all conservative triangular matrics, M the maximal group of invertible elements of Δ, B the boundary of M and . In this note little Nörlund means are located with respect to the disjoint decomposition M u B u N of Δ in terms of the zeros of the generating power series. Further, corridor matrices of finite type, that is, conservative methods with finitely many convergent diagonals, are located with respect to M u B u N.

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7

Flórez-Rueda, Roberto, and Francisco J. Moreno. "Fórmulas de direccionamiento en matrices triangularees." Revista Facultad de Ingeniería Universidad de Antioquia, no. 24 (November 29, 2001): 121–31. http://dx.doi.org/10.17533/udea.redin.326335.

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Las matrices dispersas -matrices que poseen muchos de sus elementos con valor cero-- suelen representarse en vectores con el objetivo de ahorrar espacio. Al realizar tal transformación se debe garantizar que los algoritmos desarrollados para operar con ellas ofrezcan un buen rendimiento. Se presenta a continuación un estudio de cuatro tipos de matrices dispersas triangulares, las cuales aparecen con bastante frecuencia en la práctica. Los algoritmos y las representaciones se aplican también a matrices simétricas [1]. Se incluyen los análisis y algoritmos para lograr las transformaciones deseadas, se analiza la eficiencia de cada uno de ellos y la forma como pueden mejorarse gradualmente hasta obtener algoritmos con orden de magnitud constante.

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8

Thijsse, P. "Upper Triangular Similarity of Upper Triangular Matrices." Linear Algebra and its Applications 260, no. 1-3 (1997): 119–49. http://dx.doi.org/10.1016/s0024-3795(96)00297-2.

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9

Thijsse, Philip. "Upper triangular similarity of upper triangular matrices." Linear Algebra and its Applications 260 (July 1997): 119–49. http://dx.doi.org/10.1016/s0024-3795(97)80007-9.

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10

Tsai, Ming-Cheng, Meaza Bogale, and Huajun Huang. "On triangular similarity of nilpotent triangular matrices." Linear Algebra and its Applications 596 (July 2020): 1–35. http://dx.doi.org/10.1016/j.laa.2020.02.034.

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11

Lee, K. S. Enoch. "Row-finite Matrices and Power Series Near-rings." Algebra Colloquium 17, no. 02 (2010): 301–18. http://dx.doi.org/10.1142/s1005386710000313.

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We define and investigate (structural) row-finite matrices, lower triangular matrices, and power series in near-rings by using inverse limits of some special classes of matrices. Our approach is different from that described in [8]. We show that polynomials can be embedded in power series and power series can be embedded in lower triangular matrices as in rings. We extend the concept of order of a polynomial (or a power series) from rings to near-rings. A natural topology is defined on lower triangular matrices by generalizing the concept of order of power series. We also obtain some results concerning prime and completely prime ideals in lower triangular matrix near-rings.

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12

Hołubowski, Waldemar. "Subgroups of infinite triangular matrices containing diagonal matrices." Publicationes Mathematicae Debrecen 59, no. 1-2 (2001): 45–50. http://dx.doi.org/10.5486/pmd.2001.2415.

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13

Chooi, Wai Leong, and Jinting Lau. "Derivations on rank $k$ triangular matrices." Electronic Journal of Linear Algebra 41 (January 20, 2025): 86–99. https://doi.org/10.13001/ela.2025.8953.

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Let $n\geqslant 2$ be an integer and let $T_n(\mathcal{R})$ be the algebra of $n\times n$ upper triangular matrices over a unital ring $\mathcal{R}$. In this paper, we characterize derivations $\psi:T_n(\mathcal{R})\rightarrow T_n(\mathcal{R})$ on strictly upper triangular matrices, i.e., additive maps $\psi$ satisfying $\psi(AB)=A\psi(B)+\psi(A)B$ for all strictly upper triangular matrices $A,B\in T_n(\mathcal{R})$. We then deduce this result a complete structural characterization of derivations $\psi:T_n(\mathcal{R})\rightarrow T_n(\mathcal{R})$ on rank $k$ upper triangular matrices, where $1\leqslant k\leqslant n$ is a fixed integer and $\mathcal{R}$ is a division ring.

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14

Altun, Muhammed. "Fine Spectra of Symmetric Toeplitz Operators." Abstract and Applied Analysis 2012 (2012): 1–14. http://dx.doi.org/10.1155/2012/932785.

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The fine spectra of 2-banded and 3-banded infinite Toeplitz matrices were examined by several authors. The fine spectra ofn-banded triangular Toeplitz matrices and tridiagonal symmetric matrices were computed in the following papers: Altun, “On the fine spectra of triangular toeplitz operators” (2011) and Altun, “Fine spectra of tridiagonal symmetric matrices” (2011). Here, we generalize those results to the ()-banded symmetric Toeplitz matrix operators for arbitrary positive integer .

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15

Chooi, W. L., and M. H. Lim. "Linear preservers on triangular matrices." Linear Algebra and its Applications 269, no. 1-3 (1998): 241–55. http://dx.doi.org/10.1016/s0024-3795(97)00069-4.

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16

Nagarajan, K. R., M. Paul Devasahayam, and T. Soundararajan. "Products of three triangular matrices." Linear Algebra and its Applications 292, no. 1-3 (1999): 61–71. http://dx.doi.org/10.1016/s0024-3795(99)00003-8.

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17

Guterman, A. E. "Identities of nearly triangular matrices." Sbornik: Mathematics 192, no. 6 (2001): 795–806. http://dx.doi.org/10.1070/sm2001v192n06abeh000569.

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18

Bachman, Dale, Nicholas R. Baeth, and James Gossell. "Factorizations of upper triangular matrices." Linear Algebra and its Applications 450 (June 2014): 138–57. http://dx.doi.org/10.1016/j.laa.2014.02.038.

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19

Baeth, Nicholas R., and Joel Jeffries. "Factorizations of block triangular matrices." Linear Algebra and its Applications 511 (December 2016): 403–20. http://dx.doi.org/10.1016/j.laa.2016.09.012.

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20

Hao, Pengwei. "Customizable triangular factorizations of matrices." Linear Algebra and its Applications 382 (May 2004): 135–54. http://dx.doi.org/10.1016/j.laa.2003.12.023.

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21

Vein, P. R. "Identities among certain triangular matrices." Linear Algebra and its Applications 82 (October 1986): 27–79. http://dx.doi.org/10.1016/0024-3795(86)90142-4.

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22

Benkovič, Dominik. "Jordan homomorphisms on triangular matrices." Linear and Multilinear Algebra 53, no. 5 (2005): 345–56. http://dx.doi.org/10.1080/03081080500054745.

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23

Benkovi[cbreve], Dominik. "Lie derivations on triangular matrices." Linear and Multilinear Algebra 55, no. 6 (2007): 619–26. http://dx.doi.org/10.1080/03081080701379107.

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24

CHEN, Huanyin, Orhan GÜRGÜN, and Handan KÖSE. "Uniquely strongly clean triangular matrices." TURKISH JOURNAL OF MATHEMATICS 39 (2015): 645–49. http://dx.doi.org/10.3906/mat-1408-13.

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25

Woerdeman, Hugo J. "Superoptimal completions of triangular matrices." Integral Equations and Operator Theory 20, no. 4 (1994): 491–501. http://dx.doi.org/10.1007/bf01197572.

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26

Chen, Yizhi, Xianzhong Zhao, and Zhongzhu Liu. "On upper triangular nonnegative matrices." Czechoslovak Mathematical Journal 65, no. 1 (2015): 1–20. http://dx.doi.org/10.1007/s10587-015-0158-5.

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27

Ikramov, Kh D. "Normal dilatation of triangular matrices." Mathematical Notes 60, no. 6 (1996): 649–57. http://dx.doi.org/10.1007/bf02305157.

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28

Verde-Star, Luis. "Infinite triangular matrices, q-Pascal matrices, and determinantal representations." Linear Algebra and its Applications 434, no. 1 (2011): 307–18. http://dx.doi.org/10.1016/j.laa.2010.08.022.

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29

KAMBITES, MARK. "ON THE KROHN–RHODES COMPLEXITY OF SEMIGROUPS OF UPPER TRIANGULAR MATRICES." International Journal of Algebra and Computation 17, no. 01 (2007): 187–201. http://dx.doi.org/10.1142/s0218196707003548.

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We consider the Krohn–Rhodes complexity of certain semigroups of upper triangular matrices over finite fields. We show that for any n > 1 and finite field k, the semigroups of all n × n upper triangular matrices over k and of all n × n unitriangular matrices over k have complexity n - 1. A consequence is that the complexity c > 1 of a finite semigroup places a lower bound of c + 1 on the dimension of any faithful triangular representation of that semigroup over a finite field.

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30

Teh, Wen Chean, Adrian Atanasiu, and Denis C. K. Wong. "Freeness Problem for Matrix Semigroups of Parikh Matrices." Fundamenta Informaticae 179, no. 4 (2021): 385–97. http://dx.doi.org/10.3233/fi-2021-2029.

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Since the undecidability of the mortality problem for 3 × 3 matrices over integers was proved using the Post Correspondence Problem, various studies on decision problems of matrix semigroups have emerged. The freeness problem in particular has received much attention but decidability remains open even for 2 × 2 upper triangular matrices over nonnegative integers. Parikh matrices are upper triangular matrices introduced as a generalization of Parikh vectors and have become useful tools in studying of subword occurrences. In this work, we focus on semigroups of Parikh matrices and study the freeness problem in this context.

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31

BASU, RIDDHIPRATIM, ARUP BOSE, SHIRSHENDU GANGULY, and RAJAT SUBHRA HAZRA. "SPECTRAL PROPERTIES OF RANDOM TRIANGULAR MATRICES." Random Matrices: Theory and Applications 01, no. 03 (2012): 1250003. http://dx.doi.org/10.1142/s2010326312500037.

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We prove the existence of the limiting spectral distribution (LSD) of symmetric triangular patterned matrices and also establish the joint convergence of sequences of such matrices. For the particular case of the symmetric triangular Wigner matrix, we derive expression for the moments of the LSD using properties of Catalan words. The problem of deriving explicit formulae for the moments of the LSD does not seem to be easy to solve for other patterned matrices. The LSD of the non-symmetric triangular Wigner matrix also does not seem to be easy to establish.

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32

Cornelius, Jr., E. F. "TRIANGULAR MATRICES AND COMPLETE HOMOGENEOUS SYMMETRIC POLYNOMIALS." JP Journal of Algebra, Number Theory and Applications 64, no. 4 (2025): 417–31. https://doi.org/10.17654/0972555525022.

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In [1], the authors computed the powers of the real upper triangular matrices, and demonstrated that the resulting matrices have complete homogeneous symmetric polynomials as entries. Those results are extended to infinite matrices and to infinite series of matrices over integral domains. The inverses of the are computed, as are powers of the inverses. The results also are used to produce new proofs of a famous result about complete homogeneous symmetric polynomials, without the use of generating functions.

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33

Almeida, Paulo, Miguel Beltrá, and Diego Napp. "Superregular Matrices over Finite Fields." Mathematics 13, no. 7 (2025): 1091. https://doi.org/10.3390/math13071091.

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A trivially zero minor of a matrix is a minor having all its terms in the Leibniz formula equal to zero. A matrix is superregular if all of its minors that are not trivially zero are nonzero. In the area of Coding Theory, superregular matrices over finite fields are connected with codes with optimum error correcting capabilities. There are two types of superregular matrices that yield two different types of codes. One has in all of its entries a nonzero element, and these are called full superregular matrices. The second interesting class of superregular matrices is formed by lower triangular Toeplitz matrices. In contrast to full superregular matrices, all general constructions of these matrices require very large field sizes. In this work, we investigate the construction of lower triangular Toeplitz superregular matrices over small finite prime fields. Instead of computing all possible minors, we study the structure of finite fields in order to reduce the possible nonzero minors. This allows us to restrict the huge number of possibilities that one needs to check and come up with novel constructions of superregular matrices over relatively small fields. Finally, we present concrete examples of lower triangular Toeplitz superregular matrices of sizes up to 10.

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34

Słowik, Roksana, and Driss Aiat Hadj Ahmed. "m-commuting maps on triangular and strictly triangular infinite matrices." Electronic Journal of Linear Algebra 37 (March 24, 2021): 247–55. http://dx.doi.org/10.13001/ela.2021.5083.

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Let $N_\infty(F)$ be the ring of infinite strictly upper triangular matrices with entries in an infinite field. The description of the commuting maps defined on $N_\infty(F)$, i.e. the maps $f\colon N_\infty(F)\rightarrow N_\infty(F)$ such that $[f(X),X]=0$ for every $X\in N_\infty(F)$, is presented. With the use of this result, the form of $m$-commuting maps defined on $T_\infty(F)$ -- the ring of infinite upper triangular matrices, i.e. the maps $f\colon T_\infty(F)\rightarrow T_\infty(F)$ such that $[f(X),X^m]=0$ for every $X\in T_\infty(F)$, is found.

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35

Chen, Yi-Zhi. "On Factorizations of Upper Triangular Nonnegative Matrices of Order Three." Discrete Dynamics in Nature and Society 2015 (2015): 1–6. http://dx.doi.org/10.1155/2015/960182.

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LetT3(N0)denote the semigroup of3×3upper triangular matrices with nonnegative integral-valued entries. In this paper, we investigate factorizations of upper triangular nonnegative matrices of order three. Firstly, we characterize the atoms of the subsemigroupSof the matrices inT3(N0)with nonzero determinant and give some formulas. As a consequence, problems 4a and 4c presented by Baeth et al. (2011) are each half-answered for the casen=3. And then, we consider some factorization cases of matrixAinSwithρ(A)=1and give formulas for the minimum factorization length of some special matrices inS.

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36

Zhang, Daochang, Dijana Mosic, and Predrag Stanimirovic. "Formulae for anti-triangular block matrices which include the Drazin inverse." Applicable Analysis and Discrete Mathematics, no. 00 (2024): 22. http://dx.doi.org/10.2298/aadm230418022z.

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The expressions for the Drazin inverse of two kinds of anti-triangular block matrices are developed under new and weaker assumptions relative to those already used recently in this subject. Applying our results concerning the Drazin inverse and anti-triangular block matrices, we propose some characterizations and representations of the Drazin inverse of a 2?2 block matrix. In this way, we expand some notable achievements in characterizing and representing generalized inverses of partitioned matrices.

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37

BAUER, CRAIG. "TRIANGULAR MONOIDS AND AN ANALOG TO THE DERIVED SEQUENCE OF A SOLVABLE GROUP." International Journal of Algebra and Computation 10, no. 03 (2000): 309–21. http://dx.doi.org/10.1142/s021819670000011x.

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A sequence is defined for a monoid consisting of upper triangular matrices. This sequence is analogous to the derived sequence of the solvable group. The relationship between the two sequences is investigated and sets of upper triangular partial permutation matrices, which arise naturally from this relation, are examined.

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38

Akgun, F. Aydin, and B. E. Rhoades. "Absolute Tauberian Constants for Lower Triangular Matrices." Sarajevo Journal of Mathematics 11, no. 2 (2024): 247–52. http://dx.doi.org/10.5644/sjm.11.2.10.

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In a recent paper [1] the authors obtained absolute Tauberian constants for the H-J generalized Hausdorff transformations, which generalized the corresponding results, obtained earlier by Sherif, for ordinary Hausdorff matrices. In this paper we obtain absolute Tauberian constants for regular lower triangular matrices with row sums one. As corollaries we obtain the corresponding results for factorable and weighted mean matrices. 2010 Mathematics Subject Classification. Primary: 47H10.

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39

Liu, Meili, Liwei Wang, Chun-Te Lee, and Jeng-Eng Lin. "Functions Preserving Orthogonality of Hermitian Matrices." Journal of Mathematics Research 13, no. 4 (2021): 77. http://dx.doi.org/10.5539/jmr.v13n4p77.

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Inspired by the results that functions preserve orthogonality of full matrices, upper triangular matrices, and symmetric matrices. We finish the work by finding special orthogonal matrices which satisfy the conditions of preserving orthogonality functions. We give a characterization of functions preserving orthogonality of Hermitian matrices.

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40

Chen, Huanyin, and Marjan Sheibani. "The Drazin inverse for perturbed block-operator matrices." Filomat 38, no. 7 (2024): 2311–21. https://doi.org/10.2298/fil2407311c.

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We present new formulas of Drazin inverses for anti-triangular block-operator matrices. If B?ADB = 0, B?ABD = 0 and B?ABA? = 0, the explicit representation of the Drazin inverse of a blockoperator anti-triangular matrix (A I B 0) is given. Thus a generalization of [A note on the Drazin inverse for an anti-triangular matrix, Linear Algebra Appl., 431(2009), 1910-1922] is obtained. Some applications to full block-operator matrices are thereby considered.

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41

Rhoades, B. E. "bv0-Norms for Some Triangular Matrices." Canadian Mathematical Bulletin 35, no. 3 (1992): 410–15. http://dx.doi.org/10.4153/cmb-1992-054-2.

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42

Castilho de Mello, Thiago. "Homogeneous involutions on upper triangular matrices." Archiv der Mathematik 118, no. 4 (2022): 365–74. http://dx.doi.org/10.1007/s00013-022-01712-6.

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43

Benkovič, Dominik. "Lie Triple Derivations on Triangular Matrices." Algebra Colloquium 18, spec01 (2011): 819–26. http://dx.doi.org/10.1142/s1005386711000708.

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Let [Formula: see text] be the algebra of all n × n upper triangular matrices over a commutative unital ring [Formula: see text], and let [Formula: see text] be a 2-torsion free unital [Formula: see text]-bimodule. We show that every Lie triple derivation [Formula: see text] is a sum of a standard Lie derivation and an antiderivation.

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44

Damek, E., and J. Zienkiewicz. "Affine stochastic equation with triangular matrices." Journal of Difference Equations and Applications 24, no. 4 (2018): 520–42. http://dx.doi.org/10.1080/10236198.2017.1422249.

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45

Benhida, C., E. H. Zerouali, and H. Zguitti. "Spectra of upper triangular operator matrices." Proceedings of the American Mathematical Society 133, no. 10 (2005): 3013–20. http://dx.doi.org/10.1090/s0002-9939-05-07812-3.

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46

Johnson, Charles R., Erik A. Schreiner, and Ludwig Elsner. "Eigenvalue neutrality in block triangular matrices." Linear and Multilinear Algebra 27, no. 4 (1990): 289–97. http://dx.doi.org/10.1080/03081089008818019.

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47

Yang, Yang. "Upper triangular matrices and Billiard Arrays." Linear Algebra and its Applications 493 (March 2016): 508–36. http://dx.doi.org/10.1016/j.laa.2015.12.023.

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Baeth, Nicholas R., and Rylan Sampson. "Upper triangular matrices over information algebras." Linear Algebra and its Applications 587 (February 2020): 334–57. http://dx.doi.org/10.1016/j.laa.2019.11.014.

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Cheliotis, Dimitris. "Triangular random matrices and biorthogonal ensembles." Statistics & Probability Letters 134 (March 2018): 36–44. http://dx.doi.org/10.1016/j.spl.2017.10.010.

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Lin, Fu-Rong, Wai-Ki Ching, and Michael K. Ng. "Fast inversion of triangular Toeplitz matrices." Theoretical Computer Science 315, no. 2-3 (2004): 511–23. http://dx.doi.org/10.1016/j.tcs.2004.01.005.

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