Linear Equations in Quaternions

Janovská, Drahoslava; Opfer, Gerhard

doi:10.1007/978-3-540-34288-5_94

Cited by 16 publications

(29 citation statements)

References 9 publications

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“…Of course, this fact is well-known for any linear quaternionic equation and was discussed in [11] and [3]. But to prove the theorem we have to show that: 1) the case of a hyperplane is impossible; 2) every other case is possible; 3) the last two sentences of the theorem are true; but the first of them in fact was proved in [3], so that only the second one remains to be proved.…”

Section: Sets Of Solutions Of Quaternionic Equations 421mentioning

confidence: 93%

“…We obtain more detailed information by the method of the passage to a system of real equations. The equation ax + xb = c was investigated also in Section 2 of [3] (where the authors called it Sylvester's one). We widen this information investigating possible ranks of the matrix of the corresponding system and making corresponding conclusions about the shapes.…”

Section: Introductionmentioning

confidence: 99%

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Linear Manifolds in Sets of Solutions of Quaternionic Polynomial Equations of Several Types

Mierzejewski¹

2010

Adv. Appl. Clifford Algebras

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Abstract. We give complete description of possible shapes of the set of the solutions of any quaternionic equation of the form ax + xb = c. Moreover we study the set of the solutions of a quaternionic equation of the form ax 2 + x 2 b = c by the method of sections by hyperplanes perpendicular to the real axis; for every case where such section is an unbounded linear manifold a necessary and sufficient condition is found. Mathematics Subject Classification (2010). 11R52.

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Section: Sets Of Solutions Of Quaternionic Equations 421mentioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

Linear Manifolds in Sets of Solutions of Quaternionic Polynomial Equations of Several Types

Mierzejewski¹

2010

Adv. Appl. Clifford Algebras

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“…Details for computing the matrix M are in [8,9,12]. Now, we compare the (known) real or complex eigenvalues of M with the corresponding representatives of [a].…”

Section: One Dimensional Eigenvalue Problems and Matrix Representationsmentioning

confidence: 99%

“…26 (2016) Matrices Over Nondivision Algebras 607 are considered. More recent information on problems related to quaternions and coquaternions with extensions to other algebras can be found in papers by the current authors in [8][9][10][11][12][13].…”

Section: −I II Iii −Imentioning

confidence: 99%

Matrices Over Nondivision Algebras Without Eigenvalues

Janovská

Opfer

2015

Adv. Appl. Clifford Algebras

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Abstract. We are concerned with matrices over nondivision algebras and show by an example from an R 4 algebra that these matrices do not necessarily have eigenvalues, even if these matrices are invertible. The standard condition for eigenvectors x = 0 will be replaced by the condition that x contains at least one invertible component which is the same as x = 0 for division algebras. The topic is of principal interest, and leads to the question what qualifies a matrix over a nondivision algebra to have eigenvalues. And connected with this problem is the question, whether these matrices are diagonalizable or triangulizable and allow a Schur decomposition. There is a last section where the question whether a specific matrix A has eigenvalues is extended to all eight R 4 algebras by applying numerical means. As a curiosity we found that the considered matrix A over the algebra of tessarines, which is a commutative algebra, introduced by Cockle (Phil Mag 35(3):434-437, 1849; http:// www.oocities.org/cocklebio/), possesses eigenvalues.Mathematics Subject Classification. 11E88, 15A66, 20G20, 65F15, 65H17.Keywords. Eigenvalues of matrices over nondivision algebras, Eigenvalues of matrices over coquaternions, Split-quaternions, Eigenvalues of matrices over a general R 4 algebra, Diagonalizable and triangulizable matrices over nondivision algebras.

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“…This is because lattice theory inherently relies on the commutativity in the commutative rings while quaternionic matrices or lattices inherently possess certain complexities which do not seem to be solvable [JO05]. Quaternionic matrices have been analyzed by many researches and it seems that these matrices lack many properties that matrices over an arbitrary field (commutative ring) F (R) possess.…”

Section: A Ntrū-like Cryptosystem Based On Quaternions Algebramentioning

confidence: 99%

NTRU-Like Public Key Cryptosystems beyond Dedekind Domain up to Alternative Algebra

Malekian

Zakerolhosseini

2010

Transactions on Computational Science X

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In this paper, we show that the fundamental concepts behind the Ntrū cryptosystem can be extended to a broader algebra than Dedekind domains. Also, we present an abstract and generalized algorithm for constructing a Ntrū-like cryptosystem such that the underlying algebra can be non-commutative or even non-associative.To prove the main claim, we show that it is possible to generalize Ntrū over non-commutative Quaternions (algebra in the sense of Cayley-Dikson, of dimension four over an arbitrary principal ideal domain) as well as non-associative Octonions (a power-associative and alternative algebra of dimension eight over a principal ideal domain).Given the serious challenges ahead of non-commutative/non-associative algebra in quaternionic or octonionic lattices, the proposed cryptosystems are more resistant to lattice-based attacks when compared to Ntrū.Concisely, this paper is making an abstract image of the mathematical base of Ntrū in such a way that one can make a similar cryptosystem based on various algebraic structures with the goal of better security against lattice attack and/or more capability for protocol design.

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Linear Equations in Quaternions

Cited by 16 publications

References 9 publications

Linear Manifolds in Sets of Solutions of Quaternionic Polynomial Equations of Several Types

Linear Manifolds in Sets of Solutions of Quaternionic Polynomial Equations of Several Types

Matrices Over Nondivision Algebras Without Eigenvalues

NTRU-Like Public Key Cryptosystems beyond Dedekind Domain up to Alternative Algebra

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