A generalization of the complex Autonne–Takagi factorization to quaternion matrices

The spiral-helix trajectory of the transport vehicle programmed motion in the form of a hodograph in the stationary frame of reference is considered. A relative frame of reference associated with the natural trihedral of the trajectory is introduced. The formulas of curvature and torsion of the trajectory, the unit vector of the natural trihedral, the components of the angular velocity of rotation of the natural trihedral in the proper axes and in the stationary frame of reference are set in the quaternionic matrices. The results are verified using the Frenet-Serret formulas. The mathematical apparatus of quaternion matrices is tested with the aim of adapting spatial, nonlinear problems of dynamic design of transport vehicles to a computational experiment.

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Estimates of the double determinants of quaternion matrices

Acta Mathematica Scientia ◽

10.1016/s0252-9602(10)60116-6 ◽

2010 ◽

Vol 30 (4) ◽

pp. 1189-1198

Author(s):

Feng Lianggui ◽

Cheng Wei

Keyword(s):

Quaternion Matrices

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Invariant subspaces and Jordan form

Topics in Quaternion Linear Algebra ◽

10.23943/princeton/9780691161853.003.0005 ◽

2014 ◽

Author(s):

Leiba Rodman

Keyword(s):

Existence And Uniqueness ◽

Invariant Subspaces ◽

Jordan Form ◽

Full Detail ◽

Important Class ◽

Basic Invariant ◽

Jordan Canonical Form ◽

Quaternion Matrices ◽

The Matrix ◽

The One

This chapter starts by introducing the notion of root subspaces for quaternion matrices. These are basic invariant subspaces, and the chapter proves in particular that they enjoy the Lipschitz property with respect to perturbations of the matrix. Another important class of invariant subspaces are the one-dimensional ones, i.e., generated by eigenvectors. Existence of quaternion eigenvalues and eigenvectors is proved, which leads to the Schur triangularization theorem (in the context of quaternion matrices) and its many consequences familiar for real and complex matrices. Jordan canonical form for quaternion matrices is stated and proved (both the existence and uniqueness parts) in full detail. The chapter also discusses various concepts of determinants for square-size quaternion matrices. Several applications of the Jordan form are given, including functions of matrices and boundedness properties of systems of differential and difference equations with constant quaternion coefficients.

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Some Theorems On Matrices With Real Quaternion Elements

Canadian Journal of Mathematics ◽

10.4153/cjm-1955-024-x ◽

1955 ◽

Vol 7 ◽

pp. 191-201 ◽

Cited By ~ 63

Author(s):

N. A. Wiegmann

Keyword(s):

Similarity Transformation ◽

Elementary Divisor ◽

Quaternion Matrix ◽

Unitary Similarity ◽

Triangular Form ◽

Quaternion Matrices ◽

Divisor Theory ◽

Real Quaternion ◽

Unitary Similarity Transformation

Matrices with real quaternion elements have been dealt with in earlier papers by Wolf (10) and Lee (4). In the former, an elementary divisor theory was developed for such matrices by using an isomorphism between n×n real quaternion matrices and 2n×2n matrices with complex elements. In the latter, further results were obtained (including, mainly, the transforming of a quaternion matrix into a triangular form under a unitary similarity transformation) by using a different isomorphism.

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