Fact-checked by Grok 2 weeks ago

Quaternion

In mathematics, a quaternion is a member of a noncommutative division algebra that extends the complex numbers, expressed as a linear combination a + bi + cj + dk where a, b, c, d are real numbers and i, j, k are imaginary units satisfying the relations i^2 = j^2 = k^2 = ijk = -1.^[1] This structure forms a four-dimensional vector space over the reals, with multiplication defined by the rules ij = k, jk = i, ki = j, and their anticommutative counterparts ji = -k, kj = -i, ik = -j.^[1] Quaternions were invented by Irish mathematician Sir William Rowan Hamilton on October 16, 1843, during a walk along Dublin's Royal Canal, where he realized the need for a four-dimensional extension to handle three-dimensional rotations and carved the fundamental equations into Broome Bridge.^[2] Hamilton first formally presented quaternions as quadrinomial expressions with a real scalar and an imaginary vector part in a communication to the Royal Irish Academy on November 13, 1843, marking a breakthrough in hypercomplex numbers that influenced algebra, geometry, and physics.^[3] Quaternions possess key properties that distinguish them from real and complex numbers, including associativity in multiplication but non-commutativity, the existence of inverses for nonzero elements, and the norm |q| = \sqrt{a^2 + b^2 + c^2 + d^2}, which satisfies |pq| = |p||q| for any quaternions p and q.^[1] Unit quaternions, those with norm 1, form the special unitary group SU(2), which provides a double cover of the rotation group SO(3) in three-dimensional space, enabling efficient representation of orientations via the conjugation formula v' = q v \bar{q} for a pure imaginary quaternion v.^[1] This mapping avoids issues like gimbal lock inherent in Euler angle representations and allows smooth interpolation between rotations using spherical linear interpolation (slerp).^[4] Beyond pure mathematics, quaternions have found extensive applications in fields requiring robust 3D rotation computations. In computer graphics and animation, they enable the interpolation of rotation sequences via quaternion curves, as introduced for keyframe animation, reducing computational overhead compared to 3×3 rotation matrices while preserving orthogonality.^[5] They are also employed in robotics, aerospace engineering for attitude control, and computer vision for pose estimation, where their compactness (four parameters versus nine for matrices) and resistance to numerical errors prove advantageous.^[4] Despite initial resistance due to their non-commutativity, quaternions' revival in the late 20th century underscores their enduring utility in modern computational geometry.^[5]

History

Invention and early development

Although the full quaternion algebra is attributed to the Irish mathematician William Rowan Hamilton, the French mathematician Olinde Rodrigues had independently described in 1840 a parametrization using four parameters equivalent to unit quaternions for representing rotations in three dimensions.^[6] Hamilton conceived the fundamental idea on October 16, 1843, while walking along the Royal Canal in Dublin with his wife on their way to a meeting of the Royal Irish Academy.^[7] In a moment of inspiration, Hamilton realized the non-commutative multiplication rule for the imaginary units i, j, and k, and he carved the equation i² = j² = k² = ijk = −1 into the stone of Brougham Bridge (now known as Broom Bridge) to commemorate the discovery.^[8] This event marked a breakthrough after over a decade of Hamilton's efforts to develop a mathematical framework for three-dimensional space. Hamilton's motivation stemmed from his desire to extend the algebra of complex numbers, which effectively handled two-dimensional geometry and rotations, to three dimensions in order to address geometric transformations and solve polynomial equations involving spatial quantities.^[9] He had struggled for years to define a consistent multiplication for ordered triples representing points in 3D space, analogous to complex numbers for 2D, but found that preserving key properties like a multiplicative norm required abandoning commutativity.^[10] This geometric and algebraic challenge drove Hamilton to formulate quaternions as a four-dimensional extension, incorporating a real part and three imaginary components. Hamilton first announced his discovery in a preliminary paper, "On a New Species of Imaginary Quantities connected with a Theory of Quaternions," presented to the Royal Irish Academy on November 13, 1843, and published in the Proceedings of the Royal Irish Academy, volume 2, in 1844 (pages 424–434).^[11] He elaborated on the system in a more comprehensive paper, "On Quaternions," presented on November 11, 1844, and also published in the Proceedings (volume 3).^[12] Hamilton's definitive exposition appeared in his 1853 book, Lectures on Quaternions, published by Hodges and Smith in Dublin, which systematically outlined the method based on lectures he had delivered at Trinity College since 1848.^[13] The early reception of quaternions was mixed, with significant criticism centered on their non-commutative multiplication, which challenged prevailing algebraic norms. British mathematician Augustus De Morgan, in discussions around 1841, had asserted that commutativity was an essential property of any legitimate algebra, a view that Hamilton directly contested as he developed quaternions, leading to debates that highlighted the system's unconventional nature.^[14] Despite initial skepticism, quaternions gained proponents, including Scottish physicist Peter Guthrie Tait, who extensively developed their applications in vector analysis through works like his 1867 Elementary Treatise on Quaternions and integrated them into treatments of physical problems, such as electromagnetism.^[15]

Applications in physics and mathematics

In the late 19th century, quaternions found significant applications in physics, particularly through the work of James Clerk Maxwell and Peter Guthrie Tait. Maxwell initially explored quaternions for formulating electromagnetism, corresponding with Tait on their potential to express vector operations in his electromagnetic theory, though he ultimately favored a component-based vector approach in his 1873 Treatise on Electricity and Magnetism to simplify calculations.^[16] Tait, a staunch advocate, extensively applied quaternions to electromagnetism and fluid dynamics, developing "quaternion vector analysis" as detailed in his 1867 Elementary Treatise on Quaternions, where he demonstrated their utility for describing rotational motions and strains in physical systems.^[15] Tait also extended quaternions to knot theory, inspired by Helmholtz's vortex theorems, using them to model atomic structures as knotted ether configurations in his 1885 Properties of Matter, influencing early ideas in topology and physics.^[17] By the early 20th century, quaternions declined in popularity for physical applications, supplanted by the simpler vector calculus developed independently by Josiah Willard Gibbs and Oliver Heaviside. Gibbs's 1881–1884 Elements of Vector Analysis and Heaviside's 1885–1889 Electromagnetic Theory emphasized scalar and vector products without the scalar and vector components of quaternions, making computations more straightforward for electromagnetism and mechanics, as Heaviside critiqued quaternions as overly complex for practical engineering.^[18] This shift prioritized accessibility, leading to widespread adoption of vector methods in physics textbooks and diminishing quaternion use until mid-century.^[19] Quaternions experienced a revival in the 20th century across diverse fields. In rigid body dynamics, they proved superior for representing rotations without singularities like those in Euler angles, gaining traction in aerospace engineering for attitude control systems by the 1960s, as explored in works like Altmann's 1986 Rotations, Quaternions, and Double Groups. In quantum mechanics, quaternions relate closely to Pauli matrices, which represent spin operators; the matrices σ_x, σ_y, σ_z correspond to the quaternion units i, j, k, facilitating the description of spin-1/2 particles in the Dirac and Pauli equations, as shown in Altmann's 1967 analysis linking quaternion algebra to spinor representations.^[20] Their modern resurgence in computer graphics began in the 1980s, with Ken Shoemake's 1985 SIGGRAPH paper introducing quaternion-based interpolation (slerp) for smooth 3D rotations in animation, avoiding gimbal lock and enabling efficient rendering in software like OpenGL.^[21] In pure mathematics, quaternions influenced the study of algebras, notably through Adolf Hurwitz's 1898 theorem on composition algebras, which proved that the only finite-dimensional normed division algebras over the reals are the reals (dimension 1), complexes (2), quaternions (4), and octonions (8), establishing quaternions as the highest-dimensional associative example.^[22] This result, originally published in Nachrichten der Königlichen Gesellschaft der Wissenschaften zu Göttingen, underscored quaternions' uniqueness in preserving multiplicative norms and inspired subsequent work in non-associative algebras and Lie groups.^[23]

Definition

Components and basis elements

A quaternion q is formally defined as an ordered quadruple of real numbers, expressed in the linear combination form q = a + bi + cj + [dk](/page/K), where a, b, c, d \in \mathbb{R} are the components and i, j, [k](/page/K) serve as the fundamental basis elements beyond the real unit 1, with each satisfying the square relation i^2 = j^2 = [k](/page/K)^2 = -1.^[24] This structure was introduced by William Rowan Hamilton in 1843 as an extension of complex numbers, motivated by the need to represent three-dimensional rotations algebraically.^[25] As a vector space over the real numbers \mathbb{[R](/page/R)}, the set of quaternions \mathbb{H} forms a four-dimensional space isomorphic to \mathbb{R}^4, equipped with the standard basis \{1, i, j, k\}.^[25] Unlike ordinary vectors in \mathbb{R}^4 or matrices, quaternions are distinguished by their non-commutative multiplication rule, which endows \mathbb{H} with the structure of a division algebra rather than a mere vector space.^[26] The components a, b, c, d correspond directly to the coordinates with respect to this basis, allowing quaternions to be identified with elements of \mathbb{R}^4 for addition and scalar multiplication, but requiring the specialized quaternion product for other operations. The component a is termed the scalar part of the quaternion, while bi + cj + dk constitutes the vector part, often denoted as \mathbf{v} = (b, c, d) in \mathbb{R}^3.^[25] This decomposition previews the geometric utility of quaternions, where the scalar aligns with rotational invariants and the vector encodes directional information, though full exploitation occurs through subsequent algebraic operations. Common notational conventions include boldface lettering (e.g., \mathbf{q}) to denote quaternions as distinct entities, or the paired form q = a + \mathbf{v} to emphasize the scalar-vector split, with the understanding that \mathbf{v} is a pure imaginary quaternion (zero scalar part).^[26]

Multiplication and the Hamilton product

The multiplication of quaternions is defined by extending the rules for the basis elements i, j, and k, which satisfy i^2 = j^2 = k^2 = -1, ij = k, jk = i, ki = j, and the anti-commutative relations ji = -k, kj = -i, ik = -j.^[1] These relations were introduced by William Rowan Hamilton in his foundational 1844 paper, where he established the non-commutative structure of the algebra to extend complex numbers to three dimensions.^[3] For general quaternions q_1 = s_1 + \mathbf{v}_1 and q_2 = s_2 + \mathbf{v}_2, where s_1, s_2 are scalar parts and \mathbf{v}_1, \mathbf{v}_2 are vector parts in \mathbb{R}^3, the Hamilton product is given by

q_1 q_2 = (s_1 s_2 - \mathbf{v}_1 \cdot \mathbf{v}_2) + (s_1 \mathbf{v}_2 + s_2 \mathbf{v}_1 + \mathbf{v}_1 \times \mathbf{v}_2),

combining scalar multiplication, dot products, and cross products to yield another quaternion.^[27] This formula arises from applying the basis rules distributively to the components q_1 = s_1 + x_1 i + y_1 j + z_1 k and q_2 = s_2 + x_2 i + y_2 j + z_2 k, resulting in explicit component-wise expressions such as the scalar part s_1 s_2 - x_1 x_2 - y_1 y_2 - z_1 z_2.^[28] A key feature of quaternion multiplication is its non-commutativity, as seen in the basis example ij = [k](/page/K) but ji = -[k](/page/K'), which implies ij \neq ji. To illustrate with full quaternions, consider q_1 = i = 0 + (1,0,0) and q_2 = j = 0 + (0,1,0); their product is q_1 q_2 = (0 \cdot 0 - (1,0,0) \cdot (0,1,0)) + (0 \cdot (0,1,0) + 0 \cdot (1,0,0) + (1,0,0) \times (0,1,0)) = 0 + (0,0,1) = [k](/page/K), while q_2 q_1 = -[k](/page/K').^[27] For a non-basis example, multiplying q_1 = 1 + i and q_2 = 1 + j yields q_1 q_2 = (1 \cdot 1 - (1,0,0) \cdot (0,1,0)) + (1 \cdot (0,1,0) + 1 \cdot (1,0,0) + (1,0,0) \times (0,1,0)) = 1 + (1,1,1) = 1 + i + j + [k](/page/K), but the reverse q_2 q_1 = 1 + i + j - [k](/page/K'). Quaternion multiplication is associative, meaning (q_1 q_2) q_3 = q_1 (q_2 q_3) for any quaternions q_1, q_2, q_3. This property holds due to the associativity of the basis products; for instance, (ij)k = k k = k^2 = -1 and i(jk) = i i = i^2 = -1, confirming equality on basis elements, which extends bilinearly to all quaternions.^[1]

Basic Operations

Addition, scalar multiplication, and subtraction

Quaternions form a four-dimensional vector space over the real numbers \mathbb{R}, with basis elements \{1, i, j, k\}, allowing the basic operations of addition and scalar multiplication to be performed component-wise.^[25]^[29] For two quaternions q = a + bi + cj + dk and p = e + fi + gj + hk, where a, b, c, d, e, f, g, h \in \mathbb{R}, their sum is given by

q + p = (a + e) + (b + f)i + (c + g)j + (d + h)k.

This operation is commutative and associative, mirroring the properties of vector addition in \mathbb{R}^4.^[30]^[25] Scalar multiplication by a real number r \in \mathbb{R} scales each component of the quaternion uniformly:

r q = ra + rbi + rcj + rdk.

This linearity ensures that \mathbb{H}, the set of quaternions, satisfies the axioms of a vector space over \mathbb{R}, including distributivity of scalar multiplication over vector addition.^[29]^[25] Subtraction follows as the addition of the additive inverse, where the inverse of q is -q = -a - bi - cj - dk, yielding

q - p = (a - e) + (b - f)i + (c - g)j + (d - h)k.

Thus, every quaternion has a unique additive inverse, completing the abelian group structure under addition.^[30]^[25] As an \mathbb{R}-algebra, the quaternions \mathbb{H} equip this vector space with a bilinear multiplication that distributes over addition: for quaternions q, p, s \in \mathbb{H} and r \in \mathbb{R}, q(rp + s) = rqp + qs and (rp + s)q = rpq + sq. This distributivity, alongside the non-commutative multiplication, establishes \mathbb{H} as a ring.^[29]^[25]

Conjugation, norm, and reciprocal

The conjugate of a quaternion q = a + bi + cj + dk, where a, b, c, d \in \mathbb{R} and i, j, k are the standard basis elements satisfying i^2 = j^2 = k^2 = ijk = -1, is defined as \bar{q} = a - bi - cj - dk.^[1] This operation is an involution, meaning \overline{\bar{q}} = q, and it reverses the signs of the imaginary components while preserving the real part.^[1] The norm of a quaternion q, denoted \|q\|, is the non-negative square root of the product q \bar{q}, yielding

\|q\| = \sqrt{a^2 + b^2 + c^2 + d^2} = \sqrt{q \bar{q}}.

This norm satisfies the multiplicativity property: for any quaternions q and r, \|q r\| = \|q\| \|r\|.^[1] It induces the standard Euclidean metric on the four-dimensional real vector space of quaternions, making the norm a measure of magnitude analogous to that in complex numbers.^[1] For any non-zero quaternion q, the reciprocal (or multiplicative inverse) q^{-1} exists and is given by

q^{-1} = \frac{\bar{q}}{\|q\|^2}.

This follows from the relation q \bar{q} = \|q\|^2, a real scalar, ensuring q q^{-1} = q^{-1} q = 1.^[1] Consequently, division is well-defined for non-zero quaternions via r / q = r q^{-1} for any r, highlighting the division algebra structure of the quaternions.^[1] Unit quaternions are those with norm \|q\| = [1](/page/1), satisfying q \bar{q} = [1](/page/1) and thus forming a multiplicative group under quaternion multiplication. This group, denoted \mathrm{[Sp](/page/SP)}(1), is compact and lies diffeomorphic to the 3-sphere in \mathbb{[R](/page/R)}^4.

Algebraic Properties

Division algebra structure

The quaternions \mathbb{H} form a 4-dimensional associative algebra over the real numbers \mathbb{[R](/page/R)}, with basis \{[1](/page/1), i, j, k\} and the multiplication rules i^2 = j^2 = k^2 = -[1](/page/1), ij = k, ji = -k, and cyclic permutations thereof, rendering the algebra non-commutative—for instance, ij \neq ji.^[31] This structure equips \mathbb{H} with a ring operation where addition and scalar multiplication follow vector space rules, while multiplication is bilinear and associative.^[31] As a division algebra, every non-zero quaternion q \in \mathbb{H} possesses a multiplicative inverse q^{-1} = \bar{q} / \|q\|^2, where \bar{q} denotes the conjugate (obtained by negating the vector part) and \|q\| is the Euclidean norm \sqrt{q \bar{q}}, ensuring \mathbb{H} is a division ring or skew field with unity.^[31] The absence of zero divisors follows from the multiplicativity of the norm, \|qr\| = \|q\| \|r\| for all q, r \in \mathbb{H}, which implies \|q\| > 0 if and only if q \neq 0; thus, if qr = 0, then \|q\| \|r\| = 0, so either q = 0 or r = 0.^[31] The Frobenius theorem, proved in 1877, asserts that the only finite-dimensional associative division algebras over \mathbb{R} (up to isomorphism) are \mathbb{R}, \mathbb{C}, and \mathbb{H}, with \mathbb{H} being the sole non-commutative example of dimension 4.^[32] This uniqueness underscores \mathbb{H}'s role as the non-commutative counterpart to the commutative fields \mathbb{R} and \mathbb{C}. Moreover, \mathbb{H} is a central simple algebra over \mathbb{R}, characterized by having center exactly \mathbb{R} (the scalars) and no non-trivial two-sided ideals, properties that highlight its simplicity and centrality in the Brauer group of \mathbb{R}.^[31]

Center and commutative subalgebras

The center of the quaternion algebra \mathbb{H} over the real numbers \mathbb{R}, denoted Z(\mathbb{H}), consists of all elements q \in \mathbb{H} such that q r = r q for every r \in \mathbb{H}. This center is precisely the set of scalar quaternions, isomorphic to \mathbb{R}, as only real multiples of the identity commute with all quaternions, including the basis elements i, j, k.^[33]^[25] Commutative subalgebras of \mathbb{H} arise as subspaces closed under multiplication where elements pairwise commute. Any two-dimensional commutative subalgebra containing the scalars (i.e., a copy of \mathbb{R}) is generated by the identity and a pure imaginary unit quaternion u satisfying u^2 = -1, and is thus isomorphic to the complex numbers \mathbb{C}.^[33] Such subalgebras form quadratic extensions of \mathbb{R} within \mathbb{H}, and all maximal commutative subfields of \mathbb{H} are precisely these two-dimensional ones, as higher-dimensional commutative subalgebras cannot embed in the four-dimensional division algebra \mathbb{H}.^[33] For example, the subalgebra \mathbb{R} + \mathbb{R} i = \{a + b i \mid a, b \in \mathbb{R}\} is commutative and isomorphic to \mathbb{C}, with i serving as the imaginary unit. More generally, for any angle \theta, the element u = \cos \theta \, i + \sin \theta \, j satisfies u^2 = -1 and generates another such subalgebra \mathbb{R} + \mathbb{R} u, again isomorphic to \mathbb{C}; these exhaust the maximal commutative subfields up to isomorphism.^[33]

Geometric Applications

Scalar and vector parts

A quaternion q can be decomposed into its scalar part and vector part as q = \operatorname{Re}(q) + \operatorname{Im}(q), where \operatorname{Re}(q) is a real number and \operatorname{Im}(q) is a pure quaternion representing a vector in \mathbb{R}^3.^[34]^[35] The scalar part is given by \operatorname{Re}(q) = \frac{q + \bar{q}}{2}, and the vector part by \operatorname{Im}(q) = \frac{q - \bar{q}}{2}, where \bar{q} denotes the conjugate of q.^[36] This decomposition highlights the algebraic structure of quaternions as an extension of complex numbers, with the scalar part behaving like a real component and the vector part capturing imaginary directions.^[25] The scalar and vector parts exhibit linearity under addition: for quaternions q and r, \operatorname{Im}(q + r) = \operatorname{Im}(q) + \operatorname{Im}(r) and \operatorname{Re}(q + r) = \operatorname{Re}(q) + \operatorname{Re}(r).^[34] Under multiplication, the scalar part is symmetric, satisfying \operatorname{Re}(q r) = \operatorname{Re}(r q), as it equals \operatorname{Re}(q) \operatorname{Re}(r) - \operatorname{Im}(q) \cdot \operatorname{Im}(r), where \cdot denotes the Euclidean dot product in \mathbb{R}^3.^[34] However, the vector part of the product q r does not simply equal the product of the individual vector parts, instead given by \operatorname{Re}(q) \operatorname{Im}(r) + \operatorname{Re}(r) \operatorname{Im}(q) + \operatorname{Im}(q) \times \operatorname{Im}(r), where \times is the cross product in \mathbb{R}^3.^[35] Pure quaternions, those with vanishing scalar part (\operatorname{Re}(q) = 0), form a three-dimensional real vector subspace of the quaternions isomorphic to \mathbb{R}^3.^[25] The multiplication of two pure quaternions u and v yields u v = - (u \cdot v) + (u \times v), where the scalar part is the negative dot product and the vector part is the cross product, thus embedding vector algebra within quaternion multiplication.^[34]^[25] For such pure quaternions, the dot product relates to the scalar part of their product via u \cdot v = -\operatorname{Re}(u v).^[34]

Rotations in three-dimensional space

Unit quaternions, which have norm equal to one, offer a compact and efficient way to parameterize rotations in three-dimensional Euclidean space. A rotation by an angle θ around a unit axis vector u = (u_x, u_y, u_z) is represented by the unit quaternion q = \cos\frac{\theta}{2} + \sin\frac{\theta}{2} (u_x \mathbf{i} + u_y \mathbf{j} + u_z \mathbf{k}), where i, j, and k are the standard quaternion basis elements.^[27] To apply this rotation to a vector v ∈ ℝ³, identify v as the pure quaternion (with zero scalar part) 0 + v_x i + v_y j + v_z k, and compute the rotated vector v' as the vector part of q v q⁻¹, where q⁻¹ is the conjugate of q (since |q| = 1).^[27] This conjugation action yields an explicit formula equivalent to Rodrigues' rotation formula:

\mathbf{v}' = \mathbf{v} \cos \theta + (\mathbf{u} \times \mathbf{v}) \sin \theta + \mathbf{u} (\mathbf{u} \cdot \mathbf{v}) (1 - \cos \theta)

This expression decomposes v into components parallel and perpendicular to u, rotating the perpendicular part while leaving the parallel part fixed.^[27] The mapping from the group of unit quaternions (isomorphic to SU(2)) to the rotation group SO(3) is a surjective group homomorphism known as the double cover, with kernel {1, -1}; thus, q and -q induce the same rotation, requiring a 720° rotation in quaternion space to achieve a full 360° rotation in physical space.^[37] Compared to 3×3 rotation matrices, unit quaternions use only four parameters (with one constraint) instead of nine, reducing storage and computation while avoiding singularities like gimbal lock that plague Euler angle representations.^[38] They enable smooth interpolation between rotations via spherical linear interpolation (SLERP), which traces geodesics on the unit quaternion hypersphere for natural animation paths.^[4] These properties make unit quaternions widely adopted in computer graphics for object and camera animations, as well as in robotics for attitude control and path planning.^[4]

Matrix Representations

As complex 2×2 matrices

Quaternions can be identified with a subring of the algebra of 2×2 matrices over the complex numbers via an explicit isomorphism. For a quaternion q = a + bi + cj + dk where a, b, c, d \in \mathbb{R}, the corresponding matrix is

M_q = \begin{pmatrix} a + bi & c + di \\ -c + di & a - bi \end{pmatrix}.

This mapping sends the real scalars to diagonal matrices with equal entries, the imaginary unit i to a diagonal matrix with opposite imaginary entries, and the units j and k to off-diagonal forms that capture their anticommutation with i. The representation arises from viewing \mathbb{H} as a right vector space over \mathbb{C} with basis \{1, j\}, where q = (a + bi) + (c + di)j, and applying the regular representation.^[29]^[25] The basis elements map as follows:

i \mapsto \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix}, \quad j \mapsto \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}, \quad k \mapsto \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}.

These matrices satisfy the defining relations of the quaternion algebra, such as i^2 = j^2 = k^2 = -1 and ij = k = -ji, confirming the structure preservation.^[29] Matrix multiplication corresponds exactly to quaternion multiplication under this map, establishing it as an algebra isomorphism \mathbb{H} \cong \{ M_q \mid q \in \mathbb{H} \} \subseteq M_2(\mathbb{C}). For instance, the product ij = k yields

\begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix} \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} = \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix},

matching the image of k. This embedding facilitates computations in linear algebra settings while retaining the noncommutativity of \mathbb{H}.^[25]^[29] The Euclidean norm on quaternions aligns with the matrix determinant: \det(M_q) = \|q\|^2 = a^2 + b^2 + c^2 + d^2. This follows from direct computation,

\det(M_q) = (a + bi)(a - bi) - (c + di)(-c + di) = (a^2 + b^2) + (c^2 + d^2),

providing a multiplicative property \det(M_{q_1 q_2}) = \det(M_{q_1}) \det(M_{q_2}) = \|q_1\|^2 \|q_2\|^2. Unit quaternions thus map to matrices of determinant 1.^[29] This matrix perspective enables the use of complex analysis techniques on quaternions by embedding them into M_2(\mathbb{C}), such as studying eigenvalues or traces in a holomorphic framework. It also underscores the Lie group structure, where unit quaternions form the compact group SU(2), a subgroup of SL(2,ℂ); the latter double covers the Lorentz group SO⁺(1,3) via the projective representation, linking quaternionic rotations to relativistic transformations.^[29]^[39]

As real 4×4 matrices

Quaternions can be represented as linear transformations on \mathbb{R}^4 via their regular representation, where left multiplication by a quaternion q = a + b\mathbf{i} + c\mathbf{j} + d\mathbf{k} corresponds to multiplication by a specific 4×4 real matrix acting on the column vector (w, x, y, z)^T representing the quaternion w + x\mathbf{i} + y\mathbf{j} + z\mathbf{k}.^[40] The matrix for left multiplication is given by

\begin{pmatrix} a & -b & -c & -d \\ b & a & -d & c \\ c & d & a & -b \\ d & -c & b & a \end{pmatrix}.

This representation maps the quaternion algebra to the algebra of 4×4 real matrices, preserving the non-commutative multiplication structure.^[40] For unit quaternions, where a^2 + b^2 + c^2 + d^2 = 1, the corresponding matrix is orthogonal, meaning its transpose is its inverse, and it preserves the Euclidean norm in \mathbb{R}^4.^[40] This orthogonality ensures that the transformation is an isometry, reflecting the norm-preserving property of unit quaternion multiplication.^[40] The matrix structure reveals a connection to rotations in the spatial (vector) part of \mathbb{R}^4: it can be partitioned into blocks where the action on the imaginary components (corresponding to 3D vectors) induces a rotation matrix in the 3×3 block, combined with coupling terms involving the scalar part.^[40] This real matrix representation is particularly useful in numerical computations, as it allows quaternion operations to be performed using standard real linear algebra routines, avoiding the need for complex number arithmetic and improving compatibility with real-valued software libraries.^[40]

Analytic Functions

Exponential, logarithm, and powers

The exponential function for a quaternion q is defined by the power series

\exp(q) = \sum_{n=0}^{\infty} \frac{q^n}{n!},

which converges absolutely for all q \in \mathbb{H}.^[34] This entire function extends the complex exponential to the quaternion algebra, preserving many analytic properties within suitable domains. For a quaternion expressed in polar form q = r u with r \geq 0 and \|u\| = 1, the exponential simplifies to

\exp(q) = \cos r + u \sin r.

This formula arises from the series expansion, leveraging the property that u^2 = -1 for unit pure quaternions u.^[34]^[41] The exponential map relates closely to unit quaternions, as it sends pure imaginary quaternions (those with zero real part) to the unit sphere in \mathbb{H}, analogous to how the complex exponential maps the imaginary axis to the unit circle.^[41] Specifically, for a pure unit quaternion u and real \theta, \exp(\theta u) lies on the 3-sphere S^3, parameterizing rotations in three-dimensional space.^[34] The quaternion logarithm is defined for nonzero q \in \mathbb{H} as

\log q = \log \|q\| + u \arg(q),

where u is a unit pure quaternion and \arg(q) = \acos\left( \frac{\operatorname{Re}(q)}{\|q\|} \right).^[34] This principal branch captures the magnitude via the real logarithm and the direction via the argument, but the function is multi-valued due to the non-trivial topology of the unit sphere S^3 in the quaternion space, where the exponential map is a covering map with infinitely many preimages.^[34] The branches differ by $2\pi k u for integers k, reflecting the $2\pi-periodicity along each direction.^[34] Powers of a nonzero quaternion q are defined for real exponents t using the exponential and logarithm: q^t = \exp(t \log q), accounting for the multi-valued nature of the logarithm by selecting an appropriate branch.^[41] For integer exponents n, powers can alternatively be computed via repeated multiplication q^n = q \cdot q \cdots q (n times) or, for positive integers, using the binomial theorem on the decomposition q = a + b with a real and b pure, since b^2 is real and the expansion simplifies accordingly.^[41] This approach ensures consistency with the non-commutative multiplication, though care is needed for fractional or negative powers due to branch choices.^[41]

Square roots of quaternions

Unlike the complex numbers, where every non-zero element has exactly two square roots, the non-commutative structure of the quaternions leads to varying multiplicities depending on the element. For a non-zero quaternion q, square roots always exist, but the number differs: exactly two if q is real and positive or non-real, and infinitely many if q is real and negative.^[29] For q \neq 0 that is either real and positive or non-real, the two square roots lie in the commutative subalgebra generated by q, which is isomorphic to the complex numbers. Write q = s + \mathbf{v}, where s = \operatorname{Re}(q) \in \mathbb{R} and \mathbf{v} is the vector part with \|\mathbf{v}\| = \sqrt{|q|^2 - s^2}, and |q| = \sqrt{s^2 + \|\mathbf{v}\|^2} is the norm. One square root (the principal one) is given by

\sqrt{q} = \sqrt{\frac{|q| + s}{2}} + \frac{\mathbf{v}}{\|\mathbf{v}\|} \sqrt{\frac{|q| - s}{2}},

with the other being its negative; this formula assumes |q| + s > 0, which holds unless q is real and non-positive. Equivalently, in compact form,

\sqrt{q} = \frac{|q| + q}{\sqrt{2(|q| + \operatorname{Re}(q))}}.

This explicit construction follows from solving the equation p^2 = q by decomposing into scalar and vector parts and equating components, yielding a quadratic for the scalar part of p.^[42]^[31] In general, computing square roots reduces to solving within the commutative 2-dimensional subalgebra generated by q (isomorphic to \mathbb{C}), which contains exactly two roots. For real positive q > 0, the roots are simply the real numbers \pm \sqrt{q}.^[42] A special case occurs when q is real and negative, say q = -r with r > 0. Here, |q| + s = 0, so the above formula fails, and there are infinitely many square roots: any \sqrt{r} \, \mathbf{u}, where \mathbf{u} is a pure unit quaternion (i.e., \operatorname{Re}(\mathbf{u}) = 0 and |\mathbf{u}| = 1), forming the 2-sphere S^2 in the pure quaternion subspace. In particular, the square roots of -1 are precisely the pure unit quaternions, parametrizing all rotations in 3D space.^[29]^[31] The zero quaternion has a unique square root, which is zero itself, as \mathbb{H} has no nonzero nilpotent elements. Uniqueness holds for zero, unlike the cases of real negative quaternions, reflecting the division algebra structure.^[29]

Advanced Structures

Quaternion algebras over fields

A quaternion algebra over a field F is a central simple algebra of dimension 4 over F, typically presented as the F-algebra generated by two elements i, j satisfying the relations i^2 = a, j^2 = b, and ij = -ji, where a, b \in F^\times; this algebra is denoted (a, b)_F or F\langle i, j \mid i^2 = a, j^2 = b, ij = -ji \rangle.^[29]^[25] Elements take the form x + yi + zj + wij with x, y, z, w \in F, and the algebra is non-commutative unless it splits. This structure generalizes the classical real quaternions, where the reduced norm N(x + yi + zj + wij) = x^2 - a y^2 - b z^2 + a b w^2 is a quadratic form over F.^[29] Over the real numbers \mathbb{R}, the algebra (-1, -1)_\mathbb{R} is isomorphic to the classical Hamilton quaternions \mathbb{H}, which is a division algebra since it admits no zero divisors.^[25] In contrast, over the complex numbers \mathbb{C}, every quaternion algebra is split, meaning it is isomorphic to the matrix algebra M_2(\mathbb{C}), as \mathbb{C} is algebraically closed and all central simple algebras over it split.^[29] Over finite fields \mathbb{F}_q with q odd, quaternion algebras may split or be division depending on the Hilbert symbol (a, b)_{\mathbb{F}_q}; for example, (-1, -1)_{\mathbb{F}_q} splits if q \equiv 1 \pmod{4} and is division otherwise.^[25] Quaternion algebras over a field F (of characteristic not 2) correspond precisely to the 2-torsion elements in the Brauer group \mathrm{Br}(F), where two algebras (a, b)_F and (a', b')_F are Brauer equivalent if they represent the same class, and the group operation arises from tensor products.^[29] A quaternion algebra splits (isomorphic to M_2(F)) if and only if its class is trivial in \mathrm{Br}(F); otherwise, it is a division algebra with no zero divisors. For instance, over the rationals \mathbb{Q}, quaternion algebras are division algebras precisely when their reduced norm form is anisotropic over \mathbb{Q}, meaning it has no nontrivial zero. This includes cases where the form is positive definite, such as (-1,-1)_\mathbb{Q} \cong \mathbb{H}, and indefinite forms that do not represent zero nontrivially over \mathbb{Q} but do over \mathbb{R}.^[25]

Relation to Clifford algebras and Brauer group

The real quaternions \mathbb{H} are isomorphic to the even subalgebra \mathrm{Cl}^0(3,0) of the Clifford algebra \mathrm{Cl}(3,0) over \mathbb{R}, where \mathrm{Cl}(3,0) is generated by orthogonal elements e_1, e_2, e_3 satisfying e_i^2 = 1 and e_i e_j = -e_j e_i for i \neq j. This even subalgebra consists of scalar multiples and products of even-grade elements (bivectors), capturing the non-commutative structure of quaternions within the broader framework of geometric algebras.^[43] An explicit isomorphism maps the standard quaternion basis \{1, i, j, k\} to \{1, e_2 e_3, e_3 e_1, e_1 e_2\}, where the relations i^2 = j^2 = k^2 = -1 and ij = k, jk = i, ki = j follow from the Clifford algebra multiplication rules, such as (e_2 e_3)^2 = -e_2^2 e_3^2 = -1.^[44] This embedding highlights how quaternions arise naturally as the "rotational" part of the Clifford algebra for three-dimensional Euclidean space, facilitating applications in representation theory and spinor geometry. In the context of the Brauer group \mathrm{Br}(F) of a field F (of characteristic not 2), which classifies central simple algebras up to Brauer equivalence, quaternion algebras over F generate the 2-torsion subgroup {}_2\mathrm{Br}(F). A quaternion algebra (a,b)_F is a 4-dimensional central simple algebra with basis \{1, i, j, k\} where i^2 = a, j^2 = b, k = ij = -ji, and it represents a class of order dividing 2 in \mathrm{Br}(F). Every element of order 2 in \mathrm{Br}(F) is the class of a quaternion algebra, a result tied to the structure of division algebras of exponent 2. This connection extends to Galois cohomology, where \mathrm{Br}(F) \cong H^2(\mathrm{Gal}(\overline{F}/F), \overline{F}^\times), and the 2-torsion classes correspond precisely to those arising from quaternion algebras, providing a cohomological classification of such division algebras. In algebraic topology and number theory, this framework aids in studying ramification and splitting behaviors of quaternion algebras over global fields.

Number Theory Connections

Lagrange's four-square theorem

Lagrange's four-square theorem asserts that every natural number n can be expressed as the sum of four squares of integers, that is, n = a^2 + b^2 + c^2 + d^2 for some a, b, c, d \in \mathbb{Z}.^[22] This result was established by Joseph-Louis Lagrange in 1770 through a proof involving descent on the prime factorization of n.^[45] An elegant proof using quaternions was provided by Adolf Hurwitz in 1898, leveraging the algebraic structure of the quaternion ring to demonstrate the theorem. The foundation of this approach is Euler's four-square identity, discovered by Leonhard Euler in the 1740s and communicated in a 1749 letter to Christian Goldbach, which states that the product of two sums of four squares is again a sum of four squares. Explicitly, for integers a,b,c,d,e,f,g,h,

\begin{align*} &(a^2 + b^2 + c^2 + d^2)(e^2 + f^2 + g^2 + h^2) \\ &= (ae - bf - cg - dh)^2 + (af + be + ch - dg)^2 \\ &\quad + (ag - bh + ce + df)^2 + (ah + bg - cf + de)^2. \end{align*}

This identity arises from the multiplicativity of the quaternion norm: for quaternions p = a + bi + cj + dk and q = e + fi + gj + hk, the norm N(p) = a^2 + b^2 + c^2 + d^2 satisfies N(pq) = N(p) N(q). The quaternion proof proceeds by considering the Hurwitz integers, the ring consisting of quaternions a + bi + cj + dk where a, b, c, d are all integers or all half-integers (with the property that the sum $2(a + b + c + d) is even), which forms a Euclidean domain under the norm. Since the ring is Euclidean, it is a principal ideal domain, and the norms of its elements generate all positive integers multiplicatively via the identity. To establish the theorem, it suffices to verify that every prime is a norm: 2 is represented as N(1+i) = 2, while for an odd prime p, the pigeonhole principle applied to pairs of integers modulo p yields a nontrivial quaternion with norm divisible by p, implying p itself is a norm after accounting for units of norm 1.^[45] Thus, every natural number, as a product of primes, is a product of such norms and hence a sum of four squares. Modern variants refine this using the geometry of numbers or class number computations in quaternion orders.^[22]

Quaternions as pairs of complex numbers

A quaternion q can be expressed as an ordered pair of complex numbers in the form q = z + w j, where z = a + b i, w = c + d i \in \mathbb{C}, and j satisfies the relation j z = \bar{z} j for any z \in \mathbb{C}, with the overline denoting complex conjugation.^[46] This representation embeds the standard basis elements, where i is the imaginary unit from \mathbb{C}, and j and k = i j extend the structure while preserving the quaternion algebra \mathbb{H}.^[25] The multiplication of two such quaternions is defined by the rule

(z_1 + w_1 j)(z_2 + w_2 j) = z_1 z_2 - \bar{w}_1 w_2 + (z_1 w_2 + w_1 \bar{z}_2) j,

which follows from the anticommutation properties and ensures compatibility with the standard quaternion multiplication i^2 = j^2 = k^2 = -1, i j = k = - j i.^[25] This operation makes the set of pairs (z, w) \in \mathbb{C}^2 into a non-commutative algebra isomorphic to \mathbb{H}, as verified by the embedding into $2 \times 2 complex matrices of the form

\begin{pmatrix} z & w \\ -\bar{w} & \bar{z} \end{pmatrix},

which preserves both addition and multiplication.^[46] The Euclidean norm on \mathbb{H} is preserved under this isomorphism, given by \|q\|^2 = |z|^2 + |w|^2, where | \cdot | denotes the modulus on \mathbb{C}; this corresponds to the determinant of the associated matrix and equals the square of the standard quaternion norm a^2 + b^2 + c^2 + d^2.^[25] This formulation leverages the rich structure of complex arithmetic, facilitating computations in analysis—such as holomorphic functions on quaternionic spaces—and numerical implementations in fields like computer graphics and robotics, where rotations are represented efficiently.^[25]

References

[1]
Quaternion -- from Wolfram MathWorld
The quaternions are members of a noncommutative division algebra first invented by William Rowan Hamilton. The idea for quaternions occurred to him while he ...
[2]
[PDF] ON QUATERNIONS By William Rowan Hamilton
In the theory which Sir William Hamilton submitted to the Academy in November,. 1843, the name quaternion was employed to denote a certain quadrinomial ...
[3]
Animating rotation with quaternion curves - ACM Digital Library
This paper uses quaternions to describe rotations and introduces a new spline curve on a sphere for smoothly interpolating sequences of rotations.
[4]
[PDF] Animating Rotation with Quaternion Curves
The rotations of these objects are best described using a four coordinate system, quaternions, as is shown in this paper. Of all quaternions, those on the unit ...
[5]
William Rowan Hamilton - Linda Hall Library
Aug 3, 2016 · In the letter, Hamilton revealed that he had had his quaternion moment the day before, on Oct. 16, 1843. He had been out walking with his wife, ...Missing: Brougham | Show results with:Brougham
[6]
[PDF] Twenty Years of the Hamilton Walk - MU Maths & Stats Archive site
Then, on October 16, 1843, Hamilton's mind gave birth to quaternions in a flash of inspiration, as he walked along the banks of the Royal Canal at Broombridge.
[7]
On quaternion applications in obtaining surfaces | 4open
Quaternions were advertised by Irish mathematician Sir William Rowan Hamilton in 1843. While he was searching for methods of extensioning complex numbers to ...<|separator|>
[8]
[PDF] On a new Species of Imaginary Quantities connected with a theory ...
THEORY OF QUATERNIONS. By. William Rowan Hamilton. (Proceedings of the Royal Irish Academy, 2 (1844), 424–434.) Edited by David R. Wilkins. 1999. Page 2. On a ...
[9]
On Quaternions - School of Mathematics - Trinity College Dublin
The paper On Quaternions was presented by Hamilton at a meeting of the Royal Irish Academy on November 11th, 1844, and is included in Vol. 3 of the Proceedings ...
[10]
Lectures on quaternions : containing a systematic statement of a ...
Jun 8, 2009 · A systematic statement of a new mathematical method, of which the principles were communicated in 1843 to the Royal Irish academy.
[11]
A Quaternion Journey - UC Berkeley math
For example, De Morgan posited that commutativity was essential in any algebra. But general function composition (including matrix multiplication), one of ...Missing: reception criticism
[12]
[PDF] Peter Guthrie Tait and the Promotion of Quaternions, Part I
Quaternions by reading Maxwell's Treatise and this in itself is of some moment in the history of vector analysis. And what better way for Tait to fulfil his ...
[13]
Quaternions to Vector Analysis - Kathy Loves Physics
Mar 15, 2023 · Well, it is intertwined with the history of Maxwell's equations and the life and personality of a man named Peter Tait. So, to understand, I ...
[14]
[PDF] Knots and Physics in 19th Century Scotland - EVLM
As we have mentioned, Tait's first interest in Helmholtz's paper was because he saw applications of quaternions there. It was not until 1867 that Tait verified ...
[15]
"Vectors Versus Quaternions"-The Letters in Nature - AIP Publishing
Peter Guthrie Tait was in all ways Hamilton's heir in the promotion and defense of quaternions. In the preface to the first edition of Quaternions™ he tells us ...<|control11|><|separator|>
[16]
From Maxwell's Equations and Hamilton's Quaternions to Vector ...
Mar 20, 2023 · A more concise and complete form for electromagnetism in terms of potentials appears as a subexpression of the Lorentz force.
[17]
Kinematics—Quaternions—Spinors—and Pauli's Spin Matrices
Quaternions are essentially equivalent to Pauli spin matrices which can be generated by regarding a quaternion matrix as compound—i.e., a 2×2 matrix with 2×2 ...
[18]
[PDF] SAN FRANCISCO JULY 22-26 Volume 19, Number 3, 1985
Of all quaternions, those on the unit sphere are most suitable for animation, but the question of how to construct curves on spheres has not been much explored.
[19]
[PDF] Quaternion Algebras and the Algebraic Legacy of Hamilton's ...
In 1898 A. Hurwitz proved that the only real composition algebras are R, C, H, and O. (A real composition algebra is an algebra A over. R, not necessarily ...<|control11|><|separator|>
[20]
[PDF] Composition algebras and Hurwitz' theorem on the ... - DiVA portal
In 1898 German mathematician Adolf Hurwitz presented his theorem [Hur] in which he shows that composition algebras can only exist in 1, 2, 4, or 8 dimensions, ...
[21]
[PDF] ON QUATERNIONS, OR ON A NEW SYSTEM OF IMAGINARIES IN ...
The paper On Quaternions; or on a new System of Imaginaries in Algebra, is included in The Mathematical Papers of Sir William Rowan Hamilton, vol. iii (Algebra) ...
[22]
[PDF] quaternion algebras - keith conrad
2. Hamilton's Quaternions Definition 2.1. The quaternions are H = {a + bi + cj + dk : a, b, c, d ∈ R},Missing: original | Show results with:original
[23]
[PDF] Lecture 5. Quaternions
The set of quaternions is denoted by H. This is a very concrete mathematical object. As a vector space over R, it has the standard basis 1, i, j, k.
[24]
[PDF] Quaternions and Rotations∗
Sep 10, 2013 · Quaternions are very efficient for analyzing situations where rotations in R3 are involved. A quaternion is a 4-tuple, which is a more concise ...
[25]
[PDF] Algebra and Geometry of Hamilton's Quaternions
The formula for the product of two quaternions q = (α, u = βi + γj + δk) and q = (α , u = β i + γ j + δ k) can be computed using the quaternion multiplication ...
[26]
[PDF] Quaternion algebras - John Voight
Mar 20, 2025 · Quaternion algebras sit prominently at the intersection of many mathematical subjects. They capture essential features of noncommutative ...
[27]
[PDF] ON QUATERNIONS, OR ON A NEW SYSTEM OF IMAGINARIES IN ...
The paper On Quaternions; or on a new System of Imaginaries in Algebra, is included in The Mathematical Papers of Sir William Rowan Hamilton, vol. iii (Algebra) ...
[28]
[PDF] The algebra of quaternions - Princeton University
In this chapter we introduce the quaternions and their algebra: multiplication, norm, automorphisms and antiautomorphisms, etc. We give matrix representations.
[29]
[PDF] on the frobenuis theorem on real division algebras
The Frobenius theorem states that a finite dimensional division algebra over the reals is one of the reals R, the complex numbers C or the quaternions H.
[30]
[PDF] Quaternion algebras - John Voight
Jan 10, 2024 · Quaternion algebras sit prominently at the intersection of many mathematical subjects. They capture essential features of noncommutative ...
[31]
[PDF] MATH431: Quaternions - UMD MATH
Dec 5, 2021 · Quaternions are essentially an extension of the complex numbers. Rather than introducing just one value whose square is −1 we introduce three. ...
[32]
[PDF] Quaternions and Rotations in E4 - Department of Mathematics
It is convenient to decompose a quaternion into two parts that are traditionally called its scalar and vector parts. If q = q01 + q1i + q2j + q3k, then we ...
[33]
[PDF] Quaternion Algebra and Calculus - Geometric Tools
Mar 2, 1999 · The conjugate of a quaternion is ... which selects the “real part” of the quaternion. This function satisfies the property W(q)=(q + q∗)/2.
[34]
[PDF] The Quaternions and the Spaces S3, SU(2), SO(3), and RP
thus, formally, every unit quaternion looks like a complex number cosϕ + isinϕ, except that i is replaced by a unit vector, and multiplication is quaternion ...
[35]
[PDF] CS 445 / 645 Introduction to Computer Graphics
• Remember, the axes stay in the same place during rotations. Gimbal Lock ... Biggest advantage of quaternions. • Interpolation. • Cannot linearly interpolate ...
[36]
(PDF) Quaternions, 2x2 complex matrices and Lorentz transformations
Aug 7, 2025 · We show that the matrix method to construct Lorentz transformations permit to deduce the corresponding quaternionic procedure.
[37]
[PDF] a matrix–based proof of the quaternion representation theorem for ...
rotation matrix is the product of a matrix representing left–multiplication by a unit quaternion and a matrix ... representation) or as quaternions u + xi ...
[38]
[PDF] Quaternion Algebra and Calculus
Mar 2, 1999 · The conjugate of a quaternion is ... which selects the “real part” of the quaternion. This function satisfies the property W(q)=(q + q∗)/2.<|control11|><|separator|>
[39]
https://www.researchgate.net/publication/287697540_Quaternions_2x2_complex_matrices_and_Lorentz_transformations
[40]
https://arxiv.org/pdf/math/0501249
[41]
Numerical Calculations in Clifford Algebra: A Practical Guide for ...
... quaternions P = s + p𝜎 and Q = r + q𝜎 with a basis of spatial dimensions. In this case,5 the quaternion units are I = e2 e3 , J = e3 e1 , and K = e1 e2 ...Missing: subalgebra Cl(
[42]
[PDF] Quarternions and the four square theorem - UChicago Math
The Four Square Theorem was proved by Lagrange in 1770: ev- ery positive integer is the sum of at most four squares of positive integers, i.e. n = A2 + B2 + C2 ...
[43]
[PDF] Admissibility and Frame Homotopy for Quaternionic Frames - arXiv
Instead of thinking in terms of quadruples of real numbers, it is also often convenient to interpret quaternions as pairs of complex numbers: q = a+bi+cj+dk ...<|separator|>