A quaternionic matrix is a matrix whose elements are quaternions. The quaternions form a noncommutative ring, and therefore addition and multiplication...

a quaternionic matrix representation of g assigns a square matrix of quaternions ρ(g) to each element g of G such that ρ(e) is the identity matrix and...

Quaternionic manifold – Concept in geometry Quaternionic matrix – Concept in linear algebra Quaternionic polytope – Concept in geometry Quaternionic projective...

or non-commutative field. However, it is still possible to define matrix quaternionic groups. For this reason, a vector space V is allowed to be defined...

rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix R = [...

transformations that arise by left-multiplication by some quaternionic n × n {\displaystyle n\times n} matrix, while the group S p ( 1 ) = S 3 {\displaystyle Sp(1)=S^{3}}...

ensemble. Invariant matrix ensembles are random Hermitian matrices with density on the space of real symmetric/Hermitian/quaternionic Hermitian matrices...

matrices used in mathematics, science and engineering. A matrix (plural matrices, or less commonly matrixes) is a rectangular array of numbers called entries...

}^{*}\end{bmatrix}}} Using a slightly modified (non-associative) quaternionic matrix multiplication: [ α 0 α 1 α 2 α 3 ] ∘ [ β 0 β 1 β 2 β 3 ] = [ α 0...

quasideterminant Moore, E. H. (1922), "On the determinant of an hermitian matrix with quaternionic elements. Definition and elementary properties with applications...

In mathematics, quaternionic analysis is the study of functions with quaternions as the domain and/or range. Such functions can be called functions of...

The Lie algebra of Sp(n) is given by the quaternionic skew-Hermitian matrices, the set of n-by-n quaternionic matrices that satisfy A + A † = 0 {\displaystyle...

transform is a homography used in real analysis, complex analysis, and quaternionic analysis. In the theory of Hilbert spaces, the Cayley transform is a...

projective space CPn with circles as fibers, and there are also real, quaternionic, and octonionic versions of these fibrations. In particular, the Hopf...

quaternion (in this case, the sub-field of hypercomplex analysis is called quaternionic analysis). A second instance involves functions of a motor variable where...

respect to the Riemannian metric g {\displaystyle g} and satisfy the quaternionic relations I 2 = J 2 = K 2 = I J K = − 1 {\displaystyle I^{2}=J^{2}=K^{2}=IJK=-1}...

{\displaystyle {\vec {u}}} that specifies a rotation as to axial vectors. In quaternionic formalism the choice of an orientation of the space corresponds to order...

Routines, World Scientific Publishing Co., 2012, ISBN 978-981-4425-03-2 Quaternionic Quantum Mechanics and Quantum Fields, International Series of Monographs...

quaternion; that is, a quaternion that satisfies τ2 = −1. This is the quaternionic analogue of Euler's formula. Now the unit imaginary quaternions all lie...

determinant (this is unrelated to the Moore determinant of a quaternionic Hermitian matrix). The Moore matrix has successive powers of the Frobenius automorphism...

1994 Thompson, Robert C. (1997). "The upper numerical range of a quaternionic matrix is not a complex numerical range". Linear Algebra and Its Applications...

counter-example are the special orthogonal groups in even dimension. These have the matrix − I {\displaystyle -I} in the center, and this element is path-connected...

the action of an even-graded element on a spinor is given by ordinary quaternionic multiplication. Note that the expression (1) for a vector rotation through...

matrices, and the circular symplectic ensemble (CSE) on self dual unitary quaternionic matrices. The distribution of the unitary circular ensemble CUE(n) is...

orthonormal k-frames in C n {\displaystyle \mathbb {C} ^{n}} and the quaternionic Stiefel manifold V k ( H n ) {\displaystyle V_{k}(\mathbb {H} ^{n})}...

Classical Mechanics (Second ed.). Springer. p. 225. Baez, John. "Symplectic, Quaternionic, Fermionic". Retrieved 1 February 2012. Milne, Algebraic Groups and Arithmetic...

2006) Manifolds with para-quaternionic structures are studied in differential geometry and string theory. In the para-quaternionic literature, k is replaced...

that the triple i, j and k:=ij make S into a quaternionic vector space SH. This is called a quaternionic structure. There is an invariant complex antilinear...

derivative corresponds to the integral, whence the term differintegral. In quaternionic analysis, derivatives can be defined in a similar way to real and complex...

288-cell is the only non-regular 4-polytope which is the convex hull of a quaternionic group, disregarding the infinitely many dicyclic (same as binary dihedral)...