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Journal articles on the topic 'Linear and multilinear algebra'

Author: Grafiati

Published: 4 June 2021

Last updated: 27 January 2023

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1

Qi, Liqun, Yimin Wei, Changqing Xu, and Tan Zhang. "Linear algebra and multilinear algebra." Frontiers of Mathematics in China 11, no. 3 (2016): 509–10. http://dx.doi.org/10.1007/s11464-016-0540-0.

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2

Marcus, Marvin. "Multilinear methods in linear algebra." Linear Algebra and its Applications 150 (May 1991): 41–59. http://dx.doi.org/10.1016/0024-3795(91)90158-s.

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3

Moafian, F., and Ebrahimi Vishki. "Lie higher derivations on triangular algebras revisited." Filomat 30, no. 12 (2016): 3187–94. http://dx.doi.org/10.2298/fil1612187m.

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Motivated by the extensive works of W.-S. Cheung [Linear Multilinear Algebra, 51 (2003), 299-310] and X.F. Qi [Acta Math. Sinica, English Series, 29 (2013), 1007-1018], we present the structure of Lie higher derivations on a triangular algebra explicitly. We then study those conditions under which a Lie higher derivation on a triangular algebra is proper. Our approach provides a direct proof for some known results concerning to the properness of Lie higher derivations on triangular algebras.

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4

Brusamarello, Rosali, Érica Zancanella Fornaroli, and Ednei Aparecido Santulo. "Classification of involutions on finitary incidence algebras." International Journal of Algebra and Computation 24, no. 08 (2014): 1085–98. http://dx.doi.org/10.1142/s0218196714500477.

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Let X be a connected partially ordered set and let K be a field of characteristic different from 2. We present necessary and sufficient conditions for two involutions on the finitary incidence algebra of X over K, FI (X), to be equivalent in the case when every multiplicative automorphism of FI (X) is inner. To get the classification of involutions we extend the concept of multiplicative automorphism to finitary incidence algebras and prove the Decomposition Theorem of involutions of [Anti-automorphisms and involutions on (finitary) incidence algebras, Linear Multilinear Algebra 60 (2012) 181–

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5

Taghavi, Ali, Mojtaba Nouri, Mehran Razeghi, and Vahid Darvish. "A note on non-linear ∗-Jordan derivations on ∗-algebras." Mathematica Slovaca 69, no. 3 (2019): 639–46. http://dx.doi.org/10.1515/ms-2017-0253.

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Abstract Taghavi et al. in [TAGHAVI, A.—ROHI, H.—DARVISH, V.: Non-linear ∗-Jordan derivations on von Neumann algebras, Linear Multilinear Algebra 64 (2016), 426–439] proved that the map Φ: 𝓐 → 𝓐 which satisfies the following condition $$\begin{array}{} \Phi(A\diamond B)=\Phi(A)\diamond B+A\diamond \Phi(B) \end{array} $$ where A ⋄ B = AB+BA* for every A, B ∈ 𝓐 is an additive ∗-derivation. In this short note, we prove that when A is a prime ∗-algebras and Φ: 𝓐 → 𝓐 satisfies the above condition, then Φ is ∗-additive. Moreover, if Φ(iI) is self-adjoint then Φ is derivation.

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6

Rota, Gian-Carlo. "Linear algebra and group representations, Vol. II, Multilinear algebra and group representations." Advances in Mathematics 57, no. 1 (1985): 91. http://dx.doi.org/10.1016/0001-8708(85)90107-0.

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7

Bini, Dario, Marilena Mitrouli, Marc Van Barel, and Joab Winkler. "Structured Numerical Linear and Multilinear Algebra: Analysis, Algorithms and Applications." Linear Algebra and its Applications 502 (August 2016): 1–4. http://dx.doi.org/10.1016/j.laa.2016.03.042.

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8

Chen, Huanyin, and Marjan Abdolyousefi. "Generalized Jacobson’s lemma for generalized Drazin inverses." Filomat 35, no. 7 (2021): 2267–75. http://dx.doi.org/10.2298/fil2107267c.

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We present new generalized Jacobson?s lemma for generalized Drazin inverses. This extends the main results on g-Drazin inverse of Yan, Zeng and Zhu (Linear & Multilinear Algebra, 68(2020), 81-93).

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9

Moradi, Hamid, Shigeru Furuichi, and Zahra Heydarbeygi. "New Refinement of the Operator Kantorovich Inequality." Mathematics 7, no. 2 (2019): 139. http://dx.doi.org/10.3390/math7020139.

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We focus on the improvement of operator Kantorovich type inequalities. Among the consequences, we improve the main result of the paper [H.R. Moradi, I.H. Gümüş, Z. Heydarbeygi, A glimpse at the operator Kantorovich inequality, Linear Multilinear Algebra, doi:10.1080/03081087.2018.1441799].

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10

Huang, Zhengge, and Jingjing Cui. "Improved Brauer-type eigenvalue localization sets for tensors with their applications." Filomat 34, no. 14 (2020): 4607–25. http://dx.doi.org/10.2298/fil2014607h.

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In this paper, by excluding some sets from the Brauer-type eigenvalue inclusion sets for tensors developed by Bu et al. (Linear Algebra Appl. 512 (2017) 234-248) and Li et al. (Linear and Multilinear Algebra 64 (2016) 727-736), some improved Brauer-type eigenvalue localization sets for tensors are given, which are proved to be much tighter than those put forward by Bu et al. and Li et al. As applications, some new criteria for identifying the nonsingularity of tensors are developed, which are better than some previous results. This fact is illustrated by some numerical examples.

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11

Danchev, Peter, Esther García, and Miguel Gómez Lozano. "Decompositions of matrices into potent and square-zero matrices." International Journal of Algebra and Computation 32, no. 02 (2022): 251–63. http://dx.doi.org/10.1142/s0218196722500126.

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In order to find a suitable expression of an arbitrary square matrix over an arbitrary finite commutative ring, we prove that every such matrix is always representable as a sum of a potent matrix and a nilpotent matrix of order at most two when the Jacobson radical of the ring has zero-square. This somewhat extends results of ours in Linear Multilinear Algebra (2022) established for matrices considered on arbitrary fields. Our main theorem also improves on recent results due to Abyzov et al. in Mat. Zametki (2017), Šter in Linear Algebra Appl. (2018) and Shitov in Indag. Math. (2019).

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12

Wang, Xiaoxiao, Chaoqian Li, and Yaotang Li. "A Geršgorin-type eigenvalue localization set with n parameters for stochastic matrices." Open Mathematics 16, no. 1 (2018): 298–310. http://dx.doi.org/10.1515/math-2018-0030.

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AbstractA set in the complex plane which involves n parameters in [0, 1] is given to localize all eigenvalues different from 1 for stochastic matrices. As an application of this set, an upper bound for the moduli of the subdominant eigenvalues of a stochastic matrix is obtained. Lastly, we fix n parameters in [0, 1] to give a new set including all eigenvalues different from 1, which is tighter than those provided by Shen et al. (Linear Algebra Appl. 447 (2014) 74-87) and Li et al. (Linear and Multilinear Algebra 63(11) (2015) 2159-2170) for estimating the moduli of subdominant eigenvalues.

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13

Choudhury, Projesh, Rajesh Kannan, and K. Sivakumar. "A note on linear preservers of semipositive and minimally semipositive matrices." Electronic Journal of Linear Algebra 34 (February 21, 2018): 687–94. http://dx.doi.org/10.13001/1081-3810.3864.

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Semipositive matrices (matrices that map at least one nonnegative vector to a positive vector) and minimally semipositive matrices (semipositive matrices whose no column-deleted submatrix is semipositive) are well studied in matrix theory. In this short note, the structure of linear maps which preserve the set of all semipositive/minimally semipositive matrices is studied. An open problem is solved, and some ambiguities in the article [J. Dorsey, T. Gannon, N. Jacobson, C.R. Johnson and M. Turnansky. Linear preservers of semi-positive matrices. {\em Linear and Multilinear Algebra}, 64:1853--18

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14

Danchev, Peter. "On Some Decompositions of Matrices over Algebraically Closed and Finite Fields." Journal of Siberian Federal University. Mathematics & Physics 14, no. 5 (2021): 547–53. http://dx.doi.org/10.17516/1997-1397-2021-14-5-547-553.

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We study when every square matrix over an algebraically closed field or over a finite field is decomposable into a sum of a potent matrix and a nilpotent matrix of order 2. This can be related to our recent paper, published in Linear & Multilinear Algebra (2022). We also completely address the question when each square matrix over an infinite field can be decomposed into a periodic matrix and a nilpotent matrix of order 2

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15

Zhang, Weiqian, and Chaoqian Li. "An Eigenvalue Inclusion Set for Matrices with a Constant Main Diagonal Entry." Symmetry 10, no. 12 (2018): 745. http://dx.doi.org/10.3390/sym10120745.

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A set to locate all eigenvalues for matrices with a constant main diagonal entry is given, and it is proved that this set is tighter than the well-known Geršgorin set, the Brauer set and the set proposed in (Linear and Multilinear Algebra, 60:189-199, 2012). Furthermore, by applying this result to Toeplitz matrices as a subclass of matrices with a constant main diagonal, we obtain a set including all eigenvalues of Toeplitz matrices.

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16

Huang, Shaowu, Qing-Wen Wang, Shuxia Wu, and Yaoming Yu. "Extensions of pseudo-Perron-Frobenius splitting related to generalized inverse AT,S(2)." Special Matrices 6, no. 1 (2018): 46–55. http://dx.doi.org/10.1515/spma-2018-0005.

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Abstract We in this paper define the outer-Perron-Frobenius splitting, which is an extension of the pseudo- Perron-Frobenius splitting defined in [A.N. Sushama, K. Premakumari, K.C. Sivakumar, Extensions of Perron-Frobenius splittings and relationships with nonnegative Moore-Penrose inverse, Linear and Multilinear Algebra 63 (2015) 1-11]. We present some criteria for the convergence of the outer-Perron-Frobenius splitting. The findings of this paper generalize some known results in the literatures.

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17

Heo, Jaeseong, and Maria Joiţa. "Corrigendum to ‘A Stinespring type theorem for completely positive multilinear maps on Hilbert C*-modules’ [Linear Multilinear Algebra 67 (2019), 121–140]." Linear and Multilinear Algebra 67, no. 8 (2019): 1715–16. http://dx.doi.org/10.1080/03081087.2019.1607818.

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18

Słowik, Roksana. "Corrigendum to ‘Sums of square-zero infinite matrices’ (Linear Multilinear Algebra 64 (2016), 1760–1768)." Linear and Multilinear Algebra 66, no. 6 (2017): 1277. http://dx.doi.org/10.1080/03081087.2017.1397093.

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19

Brandão, Antonio Pereira, Dimas José Gonçalves, and Plamen Koshlukov. "Graded A-identities for the matrix algebra of order two." International Journal of Algebra and Computation 26, no. 08 (2016): 1617–31. http://dx.doi.org/10.1142/s0218196716500715.

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Let [Formula: see text] be a field of characteristic 0 and let [Formula: see text]. The algebra [Formula: see text] admits a natural grading [Formula: see text] by the cyclic group [Formula: see text] of order 2. In this paper, we describe the [Formula: see text]-graded A-identities for [Formula: see text]. Recall that an A-identity for an algebra is a multilinear polynomial identity for that algebra which is a linear combination of the monomials [Formula: see text] where [Formula: see text] runs over all even permutations of [Formula: see text] that is [Formula: see text], the [Formula: see t

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20

Weiser, Alan, and Sergio E. Zarantonello. "A note on piecewise linear and multilinear table interpolation in many dimensions." Mathematics of Computation 50, no. 181 (1988): 189. http://dx.doi.org/10.1090/s0025-5718-1988-0917826-0.

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21

Wu, Yaokun, Zeying Xu, and Yinfeng Zhu. "An Expansion Property of Boolean Linear Maps." Electronic Journal of Linear Algebra 31 (February 5, 2016): 381–407. http://dx.doi.org/10.13001/1081-3810.3088.

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Given a finite set $K$, a Boolean linear map on $K$ is a map $f$ from the set $2^K$ of all subsets of $K$ into itself with $f(\emptyset )=\emptyset$ such that $f(A\cup B)=f(A)\cup f(B)$ holds for all $A,B\in 2^K$. For fixed subsets $X, Y$ of $K$, to predict if $Y$ is reachable from $X$ in the dynamical system driven by $f$, one can assume the existence of nonnegative integers $h$ with $f^h(X)=Y$, find an upper bound $\alpha$ for the minimum of all such assumed integers $h$, and test if $Y$ really appears in $f^0(X), \ldots, f^\alpha(X)$. In order to get such an upper bound estimate, this paper

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22

Berger, Guillaume O., and Raphael M. Jungers. "Worst-case topological entropy and minimal data rate for state estimation of switched linear systems." Communications of the ACM 65, no. 2 (2022): 111–18. http://dx.doi.org/10.1145/3505269.

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In this paper, we study the problem of estimating the state of a switched linear system (SLS), when the observation of the system is subject to communication constraints. We introduce the concept of worst-case topological entropy of such systems, and we show that this quantity is equal to the minimal data rate (number of bits per second) required for the state estimation of the system under arbitrary switching. Furthermore, we provide a closed-form expression for the worst-case topological entropy of switched linear systems, showing that its evaluation reduces to the computation of the joint s

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23

Khoromskij, B. N. "Structured Rank-(r1, . . . , rd) Decomposition of Function-related Tensors in R_D." Computational Methods in Applied Mathematics 6, no. 2 (2006): 194–220. http://dx.doi.org/10.2478/cmam-2006-0010.

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AbstractThe structured tensor-product approximation of multidimensional nonlocal operators by a two-level rank-(r1, . . . , rd) decomposition of related higher-order tensors is proposed and analysed. In this approach, the construction of the desired approximant to a target tensor is a reminiscence of the Tucker-type model, where the canonical components are represented in a fixed (uniform) basis, while the core tensor is given in the canonical format. As an alternative, the multilevel nested canonical decomposition is presented. The complexity analysis of the corresponding multilinear algebra

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24

Sahoo, Satyajit. "On A-numerical radius inequalities for 2 x 2 operator matrices-II." Filomat 35, no. 15 (2021): 5237–52. http://dx.doi.org/10.2298/fil2115237s.

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Rout et al. [Linear Multilinear Algebra 2020, DOI: 10.1080/03081087.2020.1810201] presented certain A-numerical radius inequalities for 2x2 operator matrices and further results on A-numerical radius of certain 2x2 operator matrices are obtained by Feki [Hacet. J. Math. Stat., 2020, DOI:10.15672/hujms.730574], very recently. The main goal of this article is to establish certain A-numerical radius equalities for operator matrices. Several new upper and lower bounds for the A-numerical radius of 2 x 2 operator matrices has been proved, where A be the 2 x 2 diagonal operator matrix whose diagonal

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25

Benzi, Michele, and Ru Huang. "Some matrix properties preserved by generalized matrix functions." Special Matrices 7, no. 1 (2019): 27–37. http://dx.doi.org/10.1515/spma-2019-0003.

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Abstract Generalized matrix functions were first introduced in [J. B. Hawkins and A. Ben-Israel, Linear and Multilinear Algebra, 1(2), 1973, pp. 163-171]. Recently, it has been recognized that these matrix functions arise in a number of applications, and various numerical methods have been proposed for their computation. The exploitation of structural properties, when present, can lead to more efficient and accurate algorithms. The main goal of this paper is to identify structural properties of matrices which are preserved by generalized matrix functions. In cases where a given property is not

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26

Banchuin, Rawid, and Roungsan Chaisrichaoren. "An Extensive Tensor Algebraic Model of Transformer." ECTI Transactions on Computer and Information Technology (ECTI-CIT) 14, no. 1 (2020): 79–91. http://dx.doi.org/10.37936/ecti-cit.2020141.153727.

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An extensive s-domain tensor algebraic model of the transformer has been proposed. Unlike the traditional matrix-vector approach which relies on the conventional linear algebra, this model which in turn assumes the multilinear algebra that is of higher dimension thus more generic, is applicable to those recently often cited transformers which often employ the unconventional characteristics i.e. frequency variant parameters, time variant parameters and fractional impedance. The examples of such transformers are the on-chip monolithic transformer, the dynamic transformers and the fractional mutu

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27

Ioppolo, Antonio. "Superalgebras with superinvolution or graded involution with colengths sequence bounded by 3." International Journal of Algebra and Computation 30, no. 04 (2020): 821–38. http://dx.doi.org/10.1142/s0218196720500204.

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Let [Formula: see text] be a superalgebra with superinvolution or graded involution over a field of characteristic zero and let [Formula: see text], [Formula: see text], be the [Formula: see text]-cocharacter of [Formula: see text]. The ∗-colengths sequence, [Formula: see text], [Formula: see text], is the sum of the multiplicities in the decomposition of the [Formula: see text]-cocharacter [Formula: see text], for all [Formula: see text]. The main purpose of this paper is to classify the superalgebras with superinvolution with ∗-colengths sequence bounded by three. Moreover, we shall extend t

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28

Pressman, Irwin S. "Generalized Commutators and the Moore-Penrose Inverse." Electronic Journal of Linear Algebra 37 (September 27, 2021): 598–612. http://dx.doi.org/10.13001/ela.2021.4991.

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This work studies the kernel of a linear operator associated with the generalized k-fold commutator. Given a set $\mathfrak{A}= \left\{ A_{1}, \ldots ,A_{k} \right\}$ of real $n \times n$ matrices, the commutator is denoted by$[A_{1}| \ldots |A_{k}]$. For a fixed set of matrices $\mathfrak{A}$ we introduce a multilinear skew-symmetric linear operator $T_{\mathfrak{A}}(X)=T(A_{1}, \ldots ,A_{k})[X]=[A_{1}| \ldots |A_{k} |X] $. For fixed $n$ and $k \ge 2n-1, \; T_{\mathfrak{A}} \equiv 0$ by the Amitsur--Levitski Theorem [2] , which motivated this work. The matrix representation $M$ of the linear

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29

Słowik, Roksana. "Corrigendum to ‘Every infinite triangular matrix is similar to a generalized infinite Jordan matrix’ (Linear Multilinear Algebra 65 (2017), 1362–1373)." Linear and Multilinear Algebra 66, no. 6 (2017): 1278. http://dx.doi.org/10.1080/03081087.2017.1397094.

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30

Choi, Yun Sung, Domingo Garcia, Sung Guen Kim, and Manuel Maestre. "THE POLYNOMIAL NUMERICAL INDEX OF A BANACH SPACE." Proceedings of the Edinburgh Mathematical Society 49, no. 1 (2006): 39–52. http://dx.doi.org/10.1017/s0013091502000810.

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AbstractIn this paper, we introduce the polynomial numerical index of order $k$ of a Banach space, generalizing to $k$-homogeneous polynomials the ‘classical’ numerical index defined by Lumer in the 1970s for linear operators. We also prove some results. Let $k$ be a positive integer. We then have the following:(i) $n^{(k)}(C(K))=1$ for every scattered compact space $K$.(ii) The inequality $n^{(k)}(E)\geq k^{k/(1-k)}$ for every complex Banach space $E$ and the constant $k^{k/(1-k)}$ is sharp.(iii) The inequalities$$ n^{(k)}(E)\leq n^{(k-1)}(E)\leq\frac{k^{(k+(1/(k-1)))}}{(k-1)^{k-1}}n^{(k)}(E)

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31

Alahmadi, A., S. Jain, Andre Leroy, and A. Sathaye. "Decompositions into products of idempotents." Electronic Journal of Linear Algebra 29 (September 20, 2015): 74–88. http://dx.doi.org/10.13001/1081-3810.2948.

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The purpose of this note is two-fold: (1) to study when quasi-Euclidean rings, regular rings and regular separative rings have the property (â) that each right (left) singular element is a product of idempotents, and (2) to consider the question: âwhen is a singular nonnegative square matrix a product of nonnegative idempotent matrices?â The importance of the class of quasi- Euclidean rings in connection with the property (â) is given by the first three authors and T.Y. Lam [Journal of Algebra, 406:154â170, 2014], where it is shown that every singular matrix over a right and left qua

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32

Panda, Swarup, and Sukanta Pati. "On the inverse of a class of weighted graphs." Electronic Journal of Linear Algebra 32 (February 6, 2017): 539–45. http://dx.doi.org/10.13001/1081-3810.3421.

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In this article, only connected bipartite graphs $G$ with a unique perfect matching $\c{M}$ are considered. Let $G_\w$ denote the weighted graph obtained from $G$ by giving weights to its edges using the positive weight function $\w:E(G)\ar (0,\ity)$ such that $\w(e)=1$ for each $e\in\c{M}$. An unweighted graph $G$ may be viewed as a weighted graph with the weight function $\w\equiv\1$ (all ones vector). A weighted graph $G_\w$ is nonsingular if its adjacency matrix $A(G_\w)$ is nonsingular. The {\em inverse} of a nonsingular weighted graph $G_\w$ is the unique weighted graph whose adjacency m

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33

Hashimoto, Mitsuyasu, and Takahiro Hayashi. "Quantum multilinear algebra." Tohoku Mathematical Journal 44, no. 4 (1992): 471–521. http://dx.doi.org/10.2748/tmj/1178227246.

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34

Dias da Silva, J. A. "Multilinear algebra: Recent applications." Linear Algebra and its Applications 241-243 (July 1996): 211–23. http://dx.doi.org/10.1016/0024-3795(95)00684-2.

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35

Duursma, Iwan, Xiao Li, and Hsin-Po Wang. "Multilinear Algebra for Distributed Storage." SIAM Journal on Applied Algebra and Geometry 5, no. 3 (2021): 552–87. http://dx.doi.org/10.1137/20m1346742.

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36

Dolotin, V. V. "Singular hypersurfaces and multilinear algebra." Journal of Mathematical Sciences 95, no. 1 (1999): 2006–27. http://dx.doi.org/10.1007/bf02169158.

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37

Brini, Andrea, Francesco Regonati, and Antonio G. B. Teolis. "Multilinear Algebra over Supersymmetric Rings." Advances in Mathematics 145, no. 1 (1999): 98–158. http://dx.doi.org/10.1006/aima.1998.1816.

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38

Johnson, R. W., C. H. Huang, and J. R. Johnson. "Multilinear algebra and parallel programming." Journal of Supercomputing 5, no. 2-3 (1991): 189–217. http://dx.doi.org/10.1007/bf00127843.

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39

Qi, Liqun, Wenyu Sun, and Yiju Wang. "Numerical multilinear algebra and its applications." Frontiers of Mathematics in China 2, no. 4 (2007): 501–26. http://dx.doi.org/10.1007/s11464-007-0031-4.

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40

Nakasho, Kazuhisa, and Yasunari Shidama. "Continuity of Multilinear Operator on Normed Linear Spaces." Formalized Mathematics 27, no. 1 (2019): 61–65. http://dx.doi.org/10.2478/forma-2019-0006.

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Summary In this article, various definitions of contuity of multilinear operators on normed linear spaces are discussed in the Mizar formalism [4], [1] and [2]. In the first chapter, several basic theorems are prepared to handle the norm of the multilinear operator, and then it is formalized that the linear space of bounded multilinear operators is a complete Banach space. In the last chapter, the continuity of the multilinear operator on finite normed spaces is addressed. Especially, it is formalized that the continuity at the origin can be extended to the continuity at every point in its who

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41

KRASILNIKOV, A. N., and D. M. RILEY. "THE TRANSFER OF A COMMUTATOR IDENTITY IN A NIL-GENERATED ALGEBRA." International Journal of Algebra and Computation 12, no. 03 (2002): 437–43. http://dx.doi.org/10.1142/s0218196702000754.

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We show that if an associative algebra over a field of characteristic 0 is generated by its nilpotent elements and satisfies a multilinear Lie commutator identity then its adjoint group satisfies the corresponding multilinear group commutator identity.

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42

Aron, Richard M., and Pablo Galindo. "Weakly compact multilinear mappings." Proceedings of the Edinburgh Mathematical Society 40, no. 1 (1997): 181–92. http://dx.doi.org/10.1017/s0013091500023543.

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The notion of Arens regularity of a bilinear form on a Banach space E is extended to continuous m-linear forms, in such a way that the natural associated linear mappings, E→L (m−1E) and (m – l)-linear mappings E × … × E → E', are all weakly compact. Among other applications, polynomials whose first derivative is weakly compact are characterized.

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43

Brualdi, Richard A., Shmuel Friedland, and Alex Pothen. "The Sparse Basis Problem and Multilinear Algebra." SIAM Journal on Matrix Analysis and Applications 16, no. 1 (1995): 1–20. http://dx.doi.org/10.1137/s0895479892230067.

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44

MERKEL, U. "A NOTE ON PARASUPERSYMMETRY ALGEBRA." Modern Physics Letters A 06, no. 03 (1991): 199–205. http://dx.doi.org/10.1142/s0217732391000166.

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Rubakov and Spiridonov have recently sketched the outlines of parasupersymmetric quantum mechanics. In particular, they showed how the bilinear part of the parasupersymmetry algebra is realized for general order of paraquantization. The remaining multilinear part is addressed in this letter.

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45

White, Neil L. "Multilinear cayley factorization." Journal of Symbolic Computation 11, no. 5-6 (1991): 421–38. http://dx.doi.org/10.1016/s0747-7171(08)80113-7.

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46

Ashdown, M. A. J., S. S. Somaroo, S. F. Gull, C. J. L. Doran, and A. N. Lasenby. "Multilinear representations of rotation groups within geometric algebra." Journal of Mathematical Physics 39, no. 3 (1998): 1566–88. http://dx.doi.org/10.1063/1.532397.

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Ahmed, Khondokar M., and Saraban Tahora. "Multilinear Algebras and Tensors with Vector Subbundle of Manifolds." Dhaka University Journal of Science 62, no. 1 (2015): 31–35. http://dx.doi.org/10.3329/dujs.v62i1.21957.

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In the present paper some aspects of tensor algebra, tensor product, exterior algebra, symmetric algebra, module of section, graded algebra, vector subbundle are studied. A Theorem 1.32. is established by using sections and fibrewise orthogonal sections of an application of Gran-Schmidt. DOI: http://dx.doi.org/10.3329/dujs.v62i1.21957 Dhaka Univ. J. Sci. 62(1): 31-35, 2014 (January)

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Nakasho, Kazuhisa. "Multilinear Operator and Its Basic Properties." Formalized Mathematics 27, no. 1 (2019): 35–45. http://dx.doi.org/10.2478/forma-2019-0004.

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Summary In the first chapter, the notion of multilinear operator on real linear spaces is discussed. The algebraic structure [2] of multilinear operators is introduced here. In the second chapter, the results of the first chapter are extended to the case of the normed spaces. This chapter shows that bounded multilinear operators on normed linear spaces constitute the algebraic structure. We referred to [3], [7], [5], [6] in this formalization.

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Oberlin, Daniel M. "A Multilinear Young's Inequality." Canadian Mathematical Bulletin 31, no. 3 (1988): 380–84. http://dx.doi.org/10.4153/cmb-1988-054-0.

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Lichtenberg, Gerwald, Georg Pangalos, Carlos Cateriano Yáñez, et al. "Implicit multilinear modeling." at - Automatisierungstechnik 70, no. 1 (2022): 13–30. http://dx.doi.org/10.1515/auto-2021-0133.

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Abstract The paper introduces a subclass of nonlinear differential-algebraic models of interest for applications. By restricting the nonlinearities to multilinear polynomials, it is possible to use modern tensor methods. This opens the door to new approximation and complexity reduction methods for large scale systems with relevant nonlinear behavior. The modeling procedures including composition, decomposition, normalization, and multilinearization steps are shown by an example of a local energy system with a nonlinear electrolyzer, a linear buck converter and a PI controller with saturation.

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