The Algebraic Riccati Equation for Quaternions

Janovská, Drahoslava; Opfer, Gerhard

doi:10.1007/s00006-013-0415-3

Cited by 6 publications

(5 citation statements)

References 14 publications

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“…are real-valued due to the inequality D b S , where D = » b 2 S + b 2 I denotes the square root of determinant norm of MV B [2]. The other two solutions, which can be obtained from (16) by substitution D → −D, are complex valued.…”

Section: Square Roots Of General MV In 3dmentioning

confidence: 99%

“…To summarize, formulas (15), (16) and (6) B in Cl 3,0 algebra in terms of radicals. Using the suggested transformations and notations, similar explicit formulas can be easily obtained for Cl 2,1 , Cl 1,2 and Cl 0,3 algebras too.…”

Section: Square Roots Of General MV In 3dmentioning

confidence: 99%

“…The matrix Riccati equation is frequently addressed in control and systems theory [1,16,17]. Solution of quaternionic (Cl 0,2 ) Riccati equation has been presented in [16].…”

Section: Application: Algebraic Riccati Equationmentioning

confidence: 99%

“…The matrix Riccati equation is frequently addressed in control and systems theory [1,16,17]. Solution of quaternionic (Cl 0,2 ) Riccati equation has been presented in [16]. Here we shall consider the Riccati-Clifford equation where matrices are replaced by MVs and matrix product by geometric (Clifford) product.…”

Section: Application: Algebraic Riccati Equationmentioning

confidence: 99%

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Square root of a multivector in 3D Clifford algebras

Dargys

Acus

2020

NAMC

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The problem of square root of multivector (MV) in real 3D (n = 3) Clifford algebras Cl 3,0, Cl 2,1, Cl 1,2 and Cl 0,3 is considered. It is shown that the square root of general 3D MV can be extracted in radicals. Also, the article presents basis-free roots of MV grades such as scalars, vectors, bivectors, pseudoscalars and their combinations, which may be useful in applied Clifford algebras. It is shown that in mentioned Clifford algebras, there appear isolated square roots and continuum of roots on hypersurfaces (infinitely many roots). Possible numerical methods to extract square root from the MV are discussed too. As an illustration, the Riccati equation formulated in terms of Clifford algebra is solved.The Clifford algebra Cl p,q is an associative noncommutative algebra, where (p, q) indicates vector space metric. In 3D case, the MV consists of the following elements (basis blades) {1, e 1 , e 2 , e 3 , e 12 , e 13 , e 23 , e 123 ≡ I}, where e i are the orthogonal basis vectors, and e ij are the bivectors (oriented planes). The last term is called the pseudoscalar. The shorthand notation, for example, e ij means e ij = e i • e j , where the circle indicates the geometric or Clifford product. Usually, the multiplication symbol is omitted, and one writes e ij = e i e j . The number of subscripts indicates the grade of basis element, so that the scalar is a grade-0 element, the vectors are the grade-1 elements etc. In the orthonormalized basis, the products of vectors satisfies e i e j + e j e i = ±2δ ij .

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Section: Square Roots Of General MV In 3dmentioning

confidence: 99%

Section: Square Roots Of General MV In 3dmentioning

confidence: 99%

“…The matrix Riccati equation is frequently addressed in control and systems theory [1,16,17]. Solution of quaternionic (Cl 0,2 ) Riccati equation has been presented in [16].…”

Section: Application: Algebraic Riccati Equationmentioning

confidence: 99%

Section: Application: Algebraic Riccati Equationmentioning

confidence: 99%

See 2 more Smart Citations

Square root of a multivector in 3D Clifford algebras

Dargys

Acus

2020

NAMC

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“…However, the equations mentioned above are instead discussed in the real number field or the complex number field. Research on the solutions of quaternion nonlinear matrix equations is uncommon [20], particularly for some structural solutions of non-integer order quaternion matrix equations, there is currently no relevant research report.…”

Section: Introductionmentioning

confidence: 99%

Hermite Positive Definite Solution of the Quaternion Matrix Equation <i>X<sup>m</sup> + B*XB = C</i>

Yao,

Liu,

Zhang

et al. 2023

JAMP

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This paper discusses the necessary and sufficient conditions for the existence of Hermite positive definite solutions of the quaternion matrix equation X m + B * XB = C (m > 0) and its iterative solution method. According to the characteristics of the coefficient matrix, a corresponding algebraic equation system is ingeniously constructed, and by discussing the equation system's solvability, the matrix equation's existence interval is obtained. Based on the characteristics of the coefficient matrix, some necessary and sufficient conditions for the existence of Hermitian positive definite solutions of the matrix equation are derived. Then, the upper and lower bounds of the positive actual solutions are estimated by using matrix inequalities. Four iteration formats are constructed according to the given conditions and existence intervals, and their convergence is proven. The selection method for the initial matrix is also provided. Finally, using the complexification operator of quaternion matrices, an equivalent iteration on the complex field is established to solve the equation in the Matlab environment. Two numerical examples are used to test the effectiveness and feasibility of the given method.

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