Matrices of quaternions

Brenner, J. L.

doi:10.2140/pjm.1951.1.329

Cited by 135 publications

(115 citation statements)

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“…We review some basic linear algebra in SL(2, H) which can be found in [2,3,9,12,14], and relate the quantities α, β, γ, δ, σ and τ to eigenvalues of quaternionic matrices. We denote the real part of a quaternion z by Re A quaternion t is a right eigenvalue of a matrix A in SL(2, H) if and only if there is a non-zero column vector v ∈ H 2 such that Av = vt. For u = 0, let w = vu −1 .…”

Section: Background Materials On Quaternionic Matricesmentioning

confidence: 99%

“…For B ∈ SL(2, H), we see from the equation BAB −1 (Bv) = Bvt that conjugate matrices have the same right eigenvalues. According to [2,Theorem 2], each quaternionic matrix is conjugate to an upper triangular matrix. This statement is equivalent to the well known fact that each (quaternionic) Möbius transformation is conjugate to a Möbius transformation that fixes ∞.…”

Section: Background Materials On Quaternionic Matricesmentioning

confidence: 99%

“…When all eigenvalues of A are similar we may or may not be able to diagonalise A. We can certainly (following Brenner, [2]) conjugate A to a triangular matrix whose diagonal entries will be similar. We now give a criterion which describes when such matrices A can be diagonalised.…”

Section: Background Materials On Quaternionic Matricesmentioning

confidence: 99%

“…According to [14 for a, b, c, d ∈ H, subject to the usual conventions regarding the point ∞ (see [2,8,14]). The map A → f from SL(2, H) to the group of quaternionic Möbius transformations is a surjective homomorphism and its kernel is the group of non-zero real scalar multiples of the identity matrix.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On the classification of quaternionic Möbius transformations

Cao¹,

Parker²,

Wang³

2004

Math. Proc. Camb. Phil. Soc.

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Section: Background Materials On Quaternionic Matricesmentioning

confidence: 99%

Section: Background Materials On Quaternionic Matricesmentioning

confidence: 99%

Section: Background Materials On Quaternionic Matricesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the classification of quaternionic Möbius transformations

Cao¹,

Parker²,

Wang³

2004

Math. Proc. Camb. Phil. Soc.

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“…The question whether standard matrices have eigenvalues does not come up, since it is known, that these matrices have always eigenvalues and the number of eigenvalues never exceeds the order n. See Horn and Johnson [7] for matrices over R, C, and Brenner [1], and Zhang [19] for matrices over H. For matrices over A the term number of eigenvalues will be made more precise a little later.…”

Section: Janovská and G Opfermentioning

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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Laplace transform: a new approach in solving linear quaternion differential equations

Cai

Kou

2017

Math Methods in App Sciences

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The theory of real quaternion differential equations has recently received more attention, but significant challenges remain the non‐commutativity structure. They have numerous applications throughout engineering and physics. In the present investigation, the Laplace transform approach to solve the linear quaternion differential equations is achieved. Specifically, the process of solving a quaternion different equation is transformed to an algebraic quaternion problem. The Laplace transform makes solving linear ODEs and the related initial value problems much easier. It has two major advantages over the methods discussed in literature. The corresponding initial value problems can be solved without first determining a general solution. More importantly, a particularly powerful feature of this method is the use of the Heaviside functions. It is helpful in solving problems, which is represented by complicated quaternion periodic functions. Copyright © 2017 John Wiley & Sons, Ltd.

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Matrices of quaternions

Cited by 135 publications

References 6 publications

On the classification of quaternionic Möbius transformations

On the classification of quaternionic Möbius transformations

Matrices Over Nondivision Algebras Without Eigenvalues

Laplace transform: a new approach in solving linear quaternion differential equations

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