On left eigenvalues of a quaternionic matrix

Huang, Liping; So, Wasin

doi:10.1016/s0024-3795(00)00246-9

Cited by 80 publications

(61 citation statements)

References 5 publications

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“…The difficulty arises from the non-commutativity of the quaternion product, which gives rise to the notion of the left and right eigenvalue decomposition [13]. Furthermore, the left eigenvalue decomposition of a quaternion is still an ongoing research topic [44]. Variants of the proposed class of QLMS algorithms are pretty much along those introduced for the LMS, this is however beyond the scope of this paper.…”

Section: Choice Of Parameters Of Qlmsmentioning

confidence: 99%

The Quaternion LMS Algorithm for Adaptive Filtering of Hypercomplex Processes

Took¹,

Mandic²

2009

IEEE Trans. Signal Process.

266

146

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Abstract-The quaternion least mean square (QLMS) algorithm is introduced for adaptive filtering of three-and fourdimensional processes, such as those observed in atmospheric modeling (wind, vector fields). These processes exhibit complex nonlinear dynamics and coupling between the dimensions, which make their component-wise processing by multiple univariate LMS, bivariate complex LMS (CLMS), or multichannel LMS (MLMS) algorithms inadequate. The QLMS accounts for these problems naturally, as it is derived directly in the quaternion domain. The analysis shows that QLMS operates inherently based on the so called "augmented" statistics, that is, both the covariance E{xx H } and pseudocovariance E{xx T } of the tap input vector x are taken into account. In addition, the operation in the quaternion domain facilitates fusion of heterogeneous data sources, for instance, the three vector dimensions of the wind field and air temperature. Simulations on both benchmark and real world data supports the approach.Index Terms-Quaternion signal processing, multidimensional adaptive filters, adaptive multi-step ahead prediction, wind modeling, data fusion via vector spaces.

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Section: Choice Of Parameters Of Qlmsmentioning

confidence: 99%

The Quaternion LMS Algorithm for Adaptive Filtering of Hypercomplex Processes

Took¹,

Mandic²

2009

IEEE Trans. Signal Process.

266

146

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“…[5]) and its fixed point(s). We say that a non-trivial element g is (i) parabolic if the norms of its right eigenvalues are 1 and it has exactly one fixed point inH,…”

Section: W-s Caomentioning

confidence: 99%

“…For a non-trivial element g, since the cardinality of its fixed point(s) and the norms of its right eigenvalues are conjugate invariant [5,6], the above classification is conjugate invariant and complete.…”

Section: W-s Caomentioning

confidence: 99%

On the Classification of Four-Dimensional Möbius Transformations

Cao

2007

Proceedings of the Edinburgh Mathematical Society

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“…So, Baker discussed right eigenvalues of the real quaternion matrices with a topological approach in [2]. On the other hand, Huang and So introduced on left eigenvalues of the real quaternion matrices [10]. After that Huang discussed consimilarity of the real quaternion matrices and obtained the Jordan canonical form of the real quaternion matrices down below consimilarity [11].…”

Section: Introductionmentioning

confidence: 99%

On the Consimilarity of Split Quaternions and Split Quaternion Matrices

Kösal

Akyiğit

Tosun

2016

Analele Universitatii "Ovidius" Constanta - Seria Matematica

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In this paper, we introduce the concept of consimilarity of split quaternions and split quaternion matrices. In this regard, we examine the solvability conditions and general solutions of the equations a x = xb and A X = XB in split quaternions and split quaternion matrices, respectively. Moreover, coneigenvalue and coneigenvector are defined for split quaternion matrices. Some consequences are also presented.

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On left eigenvalues of a quaternionic matrix

Cited by 80 publications

References 5 publications

The Quaternion LMS Algorithm for Adaptive Filtering of Hypercomplex Processes

The Quaternion LMS Algorithm for Adaptive Filtering of Hypercomplex Processes

On the Classification of Four-Dimensional Möbius Transformations

On the Consimilarity of Split Quaternions and Split Quaternion Matrices

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