Quaternion

Quaternion multiplication table

↓ × →	1	i	j	k
1	1	i	j	k
i	i	−1	k	−j
j	j	−k	−1	i
k	k	j	−i	−1
Left column shows the left factor, top row shows the right factor. Also, ${\displaystyle a\mathbf {b} =\mathbf {b} a}$ and ${\displaystyle -\mathbf {b} =(-1)\mathbf {b} }$ for ${\displaystyle a\in \mathbb {R} }$, ${\displaystyle \mathbf {b} =\mathbf {i} ,\mathbf {j} ,\mathbf {k} }$.

In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 ^{[1] [2]} and applied to mechanics in three-dimensional space. The algebra of quaternions is often denoted by H (for Hamilton), or in blackboard bold by ${\displaystyle \mathbb {H} .}$ Quaternions are not a field, because multiplication of quaternions is not, in general, commutative. Quaternions provide a definition of the quotient of two vectors in a three-dimensional space. ^{[3] [4]} Quaternions are generally represented in the form

Contents

• History
• Quaternions in physics
• Definition
• Multiplication of basis elements
• Center
• Hamilton product
• Scalar and vector parts
• Conjugation, the norm, and reciprocal
• Unit quaternion
• Algebraic properties
• Quaternions and three-dimensional geometry
• Matrix representations
• Lagrange's four-square theorem
• Quaternions as pairs of complex numbers
• Square roots
• Square roots of −1
• Square roots of arbitrary quaternions
• Functions of a quaternion variable
• Exponential, logarithm, and power functions
• Geodesic norm
• Three-dimensional and four-dimensional rotation groups
• Quaternion algebras
• Quaternions as the even part of Cl3,0(R)
• Brauer group
• Quotations
• See also
• Notes
• References
• Further reading
• Books and publications
• Links and monographs
• External links

$${\displaystyle a+b\,\mathbf {i} +c\,\mathbf {j} +d\,\mathbf {k} ,}$$

where the coefficients a, b, c, d are real numbers, and 1, i, j, k are the basis vectors or basis elements. ^[5]

Quaternions are used in pure mathematics, but also have practical uses in applied mathematics, particularly for calculations involving three-dimensional rotations, such as in three-dimensional computer graphics, computer vision, robotics, magnetic resonance imaging ^[6] and crystallographic texture analysis. ^[7] They can be used alongside other methods of rotation, such as Euler angles and rotation matrices, or as an alternative to them, depending on the application.

In modern terms, quaternions form a four-dimensional associative normed division algebra over the real numbers, and therefore a ring, also a division ring and a domain. It is a special case of a Clifford algebra, classified as ${\displaystyle \operatorname {Cl} _{0,2}(\mathbb {R} )\cong \operatorname {Cl} _{3,0}^{+}(\mathbb {R} ).}$ It was the first noncommutative division algebra to be discovered.

According to the Frobenius theorem, the algebra ${\displaystyle \mathbb {H} }$ is one of only two finite-dimensional division rings containing a proper subring isomorphic to the real numbers; the other being the complex numbers. These rings are also Euclidean Hurwitz algebras, of which the quaternions are the largest associative algebra (and hence the largest ring). Further extending the quaternions yields the non-associative octonions, which is the last normed division algebra over the real numbers. The next extension gives the sedenions, which have zero divisors and so cannot be a normed division algebra. ^[8]

The unit quaternions give a group structure on the 3-sphere S³ isomorphic to the groups Spin(3) and SU(2), i.e. the universal cover group of SO(3). The positive and negative basis vectors form the eight-element quaternion group.

History

Quaternions were introduced by Hamilton in 1843. ^[9] Important precursors to this work included Euler's four-square identity (1748) and Olinde Rodrigues' parameterization of general rotations by four parameters (1840), but neither of these writers treated the four-parameter rotations as an algebra. ^{[10] [11]} Carl Friedrich Gauss had discovered quaternions in 1819, but this work was not published until 1900. ^{[12] [13]}

Hamilton knew that the complex numbers could be interpreted as points in a plane, and he was looking for a way to do the same for points in three-dimensional space. Points in space can be represented by their coordinates, which are triples of numbers, and for many years he had known how to add and subtract triples of numbers. However, for a long time, he had been stuck on the problem of multiplication and division. He could not figure out how to calculate the quotient of the coordinates of two points in space. In fact, Ferdinand Georg Frobenius later proved in 1877 that for a division algebra over the real numbers to be finite-dimensional and associative, it cannot be three-dimensional, and there are only three such division algebras: ${\displaystyle \mathbb {R,C} }$ (complex numbers) and ${\displaystyle \mathbb {H} }$ (quaternions) which have dimension 1, 2, and 4 respectively.

The great breakthrough in quaternions finally came on Monday 16 October 1843 in Dublin, when Hamilton was on his way to the Royal Irish Academy to preside at a council meeting. As he walked along the towpath of the Royal Canal with his wife, the concepts behind quaternions were taking shape in his mind. When the answer dawned on him, Hamilton could not resist the urge to carve the formula for the quaternions,

$${\displaystyle \mathbf {i} ^{2}=\mathbf {j} ^{2}=\mathbf {k} ^{2}=\mathbf {i\,j\,k} =-1}$$

into the stone of Brougham Bridge as he paused on it. Although the carving has since faded away, there has been an annual pilgrimage since 1989 called the Hamilton Walk for scientists and mathematicians who walk from Dunsink Observatory to the Royal Canal bridge in remembrance of Hamilton's discovery.

On the following day, Hamilton wrote a letter to his friend and fellow mathematician, John T. Graves, describing the train of thought that led to his discovery. This letter was later published in a letter to the London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science ; ^[14] Hamilton states:

And here there dawned on me the notion that we must admit, in some sense, a fourth dimension of space for the purpose of calculating with triples ... An electric circuit seemed to close, and a spark flashed forth. ^[14]

Hamilton called a quadruple with these rules of multiplication a quaternion, and he devoted most of the remainder of his life to studying and teaching them. Hamilton's treatment is more geometric than the modern approach, which emphasizes quaternions' algebraic properties. He founded a school of "quaternionists", and he tried to popularize quaternions in several books. The last and longest of his books, Elements of Quaternions, ^[15] was 800 pages long; it was edited by his son and published shortly after his death.

After Hamilton's death, the Scottish mathematical physicist Peter Tait became the chief exponent of quaternions. At this time, quaternions were a mandatory examination topic in Dublin. Topics in physics and geometry that would now be described using vectors, such as kinematics in space and Maxwell's equations, were described entirely in terms of quaternions. There was even a professional research association, the Quaternion Society, devoted to the study of quaternions and other hypercomplex number systems.

From the mid-1880s, quaternions began to be displaced by vector analysis, which had been developed by Josiah Willard Gibbs, Oliver Heaviside, and Hermann von Helmholtz. Vector analysis described the same phenomena as quaternions, so it borrowed some ideas and terminology liberally from the literature on quaternions. However, vector analysis was conceptually simpler and notationally cleaner, and eventually quaternions were relegated to a minor role in mathematics and physics. A side-effect of this transition is that Hamilton's work is difficult to comprehend for many modern readers. Hamilton's original definitions are unfamiliar and his writing style was wordy and difficult to follow.

However, quaternions have had a revival since the late 20th century, primarily due to their utility in describing spatial rotations. The representations of rotations by quaternions are more compact and quicker to compute than the representations by matrices. In addition, unlike Euler angles, they are not susceptible to "gimbal lock". For this reason, quaternions are used in computer graphics, ^{[16] [17]} computer vision, robotics, ^[18] nuclear magnetic resonance image sampling, ^[6] control theory, signal processing, attitude control, physics, bioinformatics, molecular dynamics, computer simulations, and orbital mechanics. For example, it is common for the attitude control systems of spacecraft to be commanded in terms of quaternions. Quaternions have received another boost from number theory because of their relationships with the quadratic forms. ^[19]

Definition

A quaternion is an expression of the form

$${\displaystyle a+b\,\mathbf {i} +c\,\mathbf {j} +d\,\mathbf {k} ,}$$

where a, b, c, d, are real numbers, and i, j, k, are symbols that can be interpreted as unit-vectors pointing along the three spatial axes. In practice, if one of a, b, c, d is 0, the corresponding term is omitted; if a, b, c, d are all zero, the quaternion is the zero quaternion, denoted 0; if one of b, c, d equals 1, the corresponding term is written simply i, j, or k.

Hamilton describes a quaternion ${\displaystyle q=a+b\,\mathbf {i} +c\,\mathbf {j} +d\,\mathbf {k} }$, as consisting of a scalar part and a vector part. The quaternion ${\displaystyle b\,\mathbf {i} +c\,\mathbf {j} +d\,\mathbf {k} }$ is called the vector part (sometimes imaginary part) of q, and a is the scalar part (sometimes real part) of q. A quaternion that equals its real part (that is, its vector part is zero) is called a scalar or real quaternion, and is identified with the corresponding real number. That is, the real numbers are embedded in the quaternions. (More properly, the field of real numbers is isomorphic to a subset of the quaternions. The field of complex numbers is also isomorphic to three subsets of quaternions.) ^[22] A quaternion that equals its vector part is called a vector quaternion.

The set of quaternions is a 4-dimensional vector space over the real numbers, with ${\displaystyle \left\{1,\mathbf {i} ,\mathbf {j} ,\mathbf {k} \right\}}$ as a basis, by the component-wise addition

$${\displaystyle {\begin{aligned}&(a_{1}+b_{1}\mathbf {i} +c_{1}\mathbf {j} +d_{1}\mathbf {k} )+(a_{2}+b_{2}\mathbf {i} +c_{2}\mathbf {j} +d_{2}\mathbf {k} )\\[3mu]&\qquad =(a_{1}+a_{2})+(b_{1}+b_{2})\mathbf {i} +(c_{1}+c_{2})\mathbf {j} +(d_{1}+d_{2})\mathbf {k} ,\end{aligned}}}$$

and the component-wise scalar multiplication

$${\displaystyle \lambda (a+b\,\mathbf {i} +c\,\mathbf {j} +d\,\mathbf {k} )=\lambda a+(\lambda b)\mathbf {i} +(\lambda c)\mathbf {j} +(\lambda d)\mathbf {k} .}$$

A multiplicative group structure, called the Hamilton product, denoted by juxtaposition, can be defined on the quaternions in the following way:

• The real quaternion 1 is the identity element.
• The real quaternions commute with all other quaternions, that is aq = qa for every quaternion q and every real quaternion a. In algebraic terminology this is to say that the field of real quaternions are the center of this quaternion algebra.
• The product is first given for the basis elements (see next subsection), and then extended to all quaternions by using the distributive property and the center property of the real quaternions. The Hamilton product is not commutative, but is associative, thus the quaternions form an associative algebra over the real numbers.
• Additionally, every nonzero quaternion has an inverse with respect to the Hamilton product: $${\displaystyle (a+b\,\mathbf {i} +c\,\mathbf {j} +d\,\mathbf {k} )^{-1}={\frac {1}{a^{2}+b^{2}+c^{2}+d^{2}}}\,(a-b\,\mathbf {i} -c\,\mathbf {j} -d\,\mathbf {k} ).}$$

Thus the quaternions form a division algebra.

Conjugation, the norm, and reciprocal

Conjugation of quaternions is analogous to conjugation of complex numbers and to transposition (also known as reversal) of elements of Clifford algebras. To define it, let ${\displaystyle q=a+b\,\mathbf {i} +c\,\mathbf {j} +d\,\mathbf {k} }$ be a quaternion. The conjugate of q is the quaternion ${\displaystyle q^{*}=a-b\,\mathbf {i} -c\,\mathbf {j} -d\,\mathbf {k} }$. It is denoted by q^∗, q^t, ${\displaystyle {\tilde {q}}}$, or q. ^[9] Conjugation is an involution, meaning that it is its own inverse, so conjugating an element twice returns the original element. The conjugate of a product of two quaternions is the product of the conjugates in the reverse order. That is, if p and q are quaternions, then (pq)^∗ = q^∗p^∗, not p^∗q^∗.

The conjugation of a quaternion, in stark contrast to the complex setting, can be expressed with multiplication and addition of quaternions:

$${\displaystyle q^{*}=-{\tfrac {1}{2}}(q+\mathbf {i} \,q\,\mathbf {i} +\mathbf {j} \,q\,\mathbf {j} +\mathbf {k} \,q\,\mathbf {k} ).}$$

Conjugation can be used to extract the scalar and vector parts of a quaternion. The scalar part of p is ⁠1/2⁠(p + p^∗), and the vector part of p is ⁠1/2⁠(p − p^∗).

The square root of the product of a quaternion with its conjugate is called its norm and is denoted ‖q‖ (Hamilton called this quantity the tensor of q, but this conflicts with the modern meaning of "tensor"). In formulas, this is expressed as follows:

$${\displaystyle \lVert q\rVert ={\sqrt {qq^{*}}}={\sqrt {q^{*}q}}={\sqrt {a^{2}+b^{2}+c^{2}+d^{2}}}}$$

This is always a non-negative real number, and it is the same as the Euclidean norm on ${\displaystyle \mathbb {H} }$ considered as the vector space ${\displaystyle \mathbb {R} ^{4}}$. Multiplying a quaternion by a real number scales its norm by the absolute value of the number. That is, if α is real, then

$${\displaystyle \lVert \alpha q\rVert =\left|\alpha \right|\,\lVert q\rVert .}$$

This is a special case of the fact that the norm is multiplicative, meaning that

$${\displaystyle \lVert pq\rVert =\lVert p\rVert \,\lVert q\rVert }$$

for any two quaternions p and q. Multiplicativity is a consequence of the formula for the conjugate of a product. Alternatively it follows from the identity

$${\displaystyle \det {\begin{pmatrix}a+ib&id+c\\id-c&a-ib\end{pmatrix}}=a^{2}+b^{2}+c^{2}+d^{2},}$$

(where i denotes the usual imaginary unit) and hence from the multiplicative property of determinants of square matrices.

This norm makes it possible to define the distanced(p, q) between p and q as the norm of their difference:

$${\displaystyle d(p,q)=\lVert p-q\rVert .}$$

This makes ${\displaystyle \mathbb {H} }$ a metric space. Addition and multiplication are continuous in regard to the associated metric topology. This follows with exactly the same proof as for the real numbers ${\displaystyle \mathbb {R} }$ from the fact that ${\displaystyle \mathbb {H} }$ is a normed algebra.

Algebraic properties

The set ${\displaystyle \mathbb {H} }$ of all quaternions is a vector space over the real numbers with dimension  4. ^[c] Multiplication of quaternions is associative and distributes over vector addition, but with the exception of the scalar subset, it is not commutative. Therefore, the quaternions ${\displaystyle \mathbb {H} }$ are a non-commutative, associative algebra over the real numbers. Even though ${\displaystyle \mathbb {H} }$ contains copies of the complex numbers, it is not an associative algebra over the complex numbers.

Because it is possible to divide quaternions, they form a division algebra. This is a structure similar to a field except for the non-commutativity of multiplication. Finite-dimensional associative division algebras over the real numbers are very rare. The Frobenius theorem states that there are exactly three: ${\displaystyle \mathbb {R} }$, ${\displaystyle \mathbb {C} }$, and ${\displaystyle \mathbb {H} }$. The norm makes the quaternions into a normed algebra, and normed division algebras over the real numbers are also very rare: Hurwitz's theorem says that there are only four: ${\displaystyle \mathbb {R} }$, ${\displaystyle \mathbb {C} }$, ${\displaystyle \mathbb {H} }$, and ${\displaystyle \mathbb {O} }$ (the octonions). The quaternions are also an example of a composition algebra and of a unital Banach algebra.

Because the product of any two basis vectors is plus or minus another basis vector, the set {±1, ±i, ±j, ±k} forms a group under multiplication. This non-abelian group is called the quaternion group and is denoted Q₈. ^[26] The real group ring of Q₈ is a ring ${\displaystyle \mathbb {R} [\mathrm {Q} _{8}]}$ which is also an eight-dimensional vector space over ${\displaystyle \mathbb {R} .}$ It has one basis vector for each element of ${\displaystyle \mathrm {Q} _{8}.}$ The quaternions are isomorphic to the quotient ring of ${\displaystyle \mathbb {R} [\mathrm {Q} _{8}]}$ by the ideal generated by the elements 1 + (−1), i + (−i), j + (−j), and k + (−k). Here the first term in each of the differences is one of the basis elements 1, i, j, and k, and the second term is one of basis elements −1, −i, −j, and −k, not the additive inverses of 1, i, j, and k.

Quaternions and three-dimensional geometry

The vector part of a quaternion can be interpreted as a coordinate vector in ${\displaystyle \mathbb {R} ^{3};}$ therefore, the algebraic operations of the quaternions reflect the geometry of ${\displaystyle \mathbb {R} ^{3}.}$ Operations such as the vector dot and cross products can be defined in terms of quaternions, and this makes it possible to apply quaternion techniques wherever spatial vectors arise. A useful application of quaternions has been to interpolate the orientations of key-frames in computer graphics. ^[16]

For the remainder of this section, i, j, and k will denote both the three imaginary ^[27] basis vectors of ${\displaystyle \mathbb {H} }$ and a basis for ${\displaystyle \mathbb {R} ^{3}.}$ Replacing i by −i, j by −j, and k by −k sends a vector to its additive inverse, so the additive inverse of a vector is the same as its conjugate as a quaternion. For this reason, conjugation is sometimes called the spatial inverse.

For two vector quaternions p = b₁i + c₁j + d₁k and q = b₂i + c₂j + d₂k their dot product, by analogy to vectors in ${\displaystyle \mathbb {R} ^{3},}$ is

$${\displaystyle p\cdot q=b_{1}b_{2}+c_{1}c_{2}+d_{1}d_{2}.}$$

It can also be expressed in a component-free manner as

$${\displaystyle p\cdot q=\textstyle {\frac {1}{2}}(p^{*}q+q^{*}p)=\textstyle {\frac {1}{2}}(pq^{*}+qp^{*}).}$$

This is equal to the scalar parts of the products pq^∗, qp^∗, p^∗q, and q^∗p. Note that their vector parts are different.

The cross product of p and q relative to the orientation determined by the ordered basis i, j, and k is

$${\displaystyle p\times q=(c_{1}d_{2}-d_{1}c_{2})\mathbf {i} +(d_{1}b_{2}-b_{1}d_{2})\mathbf {j} +(b_{1}c_{2}-c_{1}b_{2})\mathbf {k} .}$$

(Recall that the orientation is necessary to determine the sign.) This is equal to the vector part of the product pq (as quaternions), as well as the vector part of −q^∗p^∗. It also has the formula

$${\displaystyle p\times q=\textstyle {\tfrac {1}{2}}(pq-qp).}$$

For the commutator, [p, q] = pq − qp, of two vector quaternions one obtains

$${\displaystyle [p,q]=2p\times q.}$$

In general, let p and q be quaternions and write

$${\displaystyle {\begin{aligned}p&=p_{\text{s}}+p_{\text{v}},\\[5mu]q&=q_{\text{s}}+q_{\text{v}},\end{aligned}}}$$

where p_s and q_s are the scalar parts, and p_v and q_v are the vector parts of p and q. Then we have the formula

$${\displaystyle pq=(pq)_{\text{s}}+(pq)_{\text{v}}=(p_{\text{s}}q_{\text{s}}-p_{\text{v}}\cdot q_{\text{v}})+(p_{\text{s}}q_{\text{v}}+q_{\text{s}}p_{\text{v}}+p_{\text{v}}\times q_{\text{v}}).}$$

This shows that the noncommutativity of quaternion multiplication comes from the multiplication of vector quaternions. It also shows that two quaternions commute if and only if their vector parts are collinear. Hamilton ^[28] showed that this product computes the third vertex of a spherical triangle from two given vertices and their associated arc-lengths, which is also an algebra of points in Elliptic geometry.

Unit quaternions can be identified with rotations in ${\displaystyle \mathbb {R} ^{3}}$ and were called versors by Hamilton. ^[28] Also see Quaternions and spatial rotation for more information about modeling three-dimensional rotations using quaternions.

See Hanson (2005) ^[29] for visualization of quaternions.

Matrix representations

Just as complex numbers can be represented as matrices, so can quaternions. There are at least two ways of representing quaternions as matrices in such a way that quaternion addition and multiplication correspond to matrix addition and matrix multiplication. One is to use 2 × 2 complex matrices, and the other is to use 4 × 4 real matrices. In each case, the representation given is one of a family of linearly related representations. These are injective homomorphisms from ${\displaystyle \mathbb {H} }$ to the matrix rings M(2,C) and M(4,R), respectively.

The quaternion a + bi + cj + dk can be represented using a 2 × 2 complex matrix as

$${\displaystyle {\begin{bmatrix}{\phantom {-}}a+bi&c+di\\-c+di&a-bi\end{bmatrix}}.}$$

This representation has the following properties:

• Constraining any two of b, c and d to zero produces a representation of complex numbers. For example, setting c = d = 0 produces a diagonal complex matrix representation of complex numbers, and setting b = d = 0 produces a real matrix representation.
• The norm of a quaternion (the square root of the product with its conjugate, as with complex numbers) is the square root of the determinant of the corresponding matrix. ^[30]
• The scalar part of a quaternion is one half of the matrix trace.
• The conjugate of a quaternion corresponds to the conjugate transpose of the matrix.
• By restriction this representation yields an isomorphism between the subgroup of unit quaternions and their image SU(2). Topologically, the unit quaternions are the 3-sphere, so the underlying space of SU(2) is also a 3-sphere. The group SU(2) is important for describing spin in quantum mechanics; see Pauli matrices.
• There is a strong relation between quaternion units and Pauli matrices. The 2 × 2 complex matrix above can be written as ${\displaystyle aI+bi\sigma _{3}+ci\sigma _{2}+di\sigma _{1}}$, so in this representation the quaternion units {±1, ±i, ±j, ±k} correspond to {±I, ${\displaystyle \pm i\sigma _{3}}$,${\displaystyle \pm i\sigma _{2}}$, ${\displaystyle \pm i\sigma _{1}}$} = {±I, ${\displaystyle \pm \sigma _{1}\sigma _{2}}$,${\displaystyle \pm \sigma _{3}\sigma _{1}}$, ${\displaystyle \pm \sigma _{2}\sigma _{3}}$}, so multiplying any two Pauli matrices always yields a quaternion unit matrix, all of them except for −1. One obtains −1 via i² = j² = k² = i j k = −1; e.g. the last equality is $${\displaystyle \mathbf {i\;j\;k} =\sigma _{1}\sigma _{2}\sigma _{3}\sigma _{1}\sigma _{2}\sigma _{3}=-1.}$$
• The representation in M(2,C) is not unique. A different convention, that preserves the direction of cyclic ordering between the quaternions and the Pauli matrices, is to choose $${\displaystyle 1\mapsto \mathbf {I} ,\quad \mathbf {i} \mapsto -i\sigma _{1}=-\sigma _{2}\sigma _{3},\quad \mathbf {j} \mapsto -i\sigma _{2}=-\sigma _{3}\sigma _{1},\quad \mathbf {k} \mapsto -i\sigma _{3}=-\sigma _{1}\sigma _{2},}$$
  This gives an alternative representation, ^[31] $${\displaystyle a+b\,\mathbf {i} +c\,\mathbf {j} +d\,\mathbf {k} \mapsto {\begin{bmatrix}a-di&-c-bi\\c-bi&{\phantom {-}}a+di\end{bmatrix}}.}$$

Using 4 × 4 real matrices, that same quaternion can be written as

$${\displaystyle {\begin{aligned}\left[{\begin{array}{rrrr}a&-b&-c&-d\\b&a&-d&c\\c&d&a&-b\\d&-c&b&a\end{array}}\right]&=a\left[{\begin{array}{rrrr}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{array}}\right]+b\left[{\begin{array}{rrrr}0&-1&0&0\\1&0&0&0\\0&0&0&-1\\0&0&1&0\end{array}}\right]\\[10mu]&\qquad +c\left[{\begin{array}{rrrr}0&0&-1&0\\0&0&0&1\\1&0&0&0\\0&-1&0&0\end{array}}\right]+d\left[{\begin{array}{rrrr}0&0&0&-1\\0&0&-1&0\\0&1&0&0\\1&0&0&0\end{array}}\right].\end{aligned}}}$$

However, the representation of quaternions in M(4,R) is not unique. For example, the same quaternion can also be represented as

$${\displaystyle {\begin{aligned}\left[{\begin{array}{rrrr}a&d&-b&-c\\-d&a&c&-b\\b&-c&a&-d\\c&b&d&a\end{array}}\right]&=a\left[{\begin{array}{rrrr}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{array}}\right]+b\left[{\begin{array}{rrrr}0&0&-1&0\\0&0&0&-1\\1&0&0&0\\0&1&0&0\end{array}}\right]\\[10mu]&\qquad +c\left[{\begin{array}{rrrr}0&0&0&-1\\0&0&1&0\\0&-1&0&0\\1&0&0&0\end{array}}\right]+d\left[{\begin{array}{rrrr}0&1&0&0\\-1&0&0&0\\0&0&0&-1\\0&0&1&0\end{array}}\right].\end{aligned}}}$$

There exist 48 distinct matrix representations of this form in which one of the matrices represents the scalar part and the other three are all skew-symmetric. More precisely, there are 48 sets of quadruples of matrices with these symmetry constraints such that a function sending 1, i, j, and k to the matrices in the quadruple is a homomorphism, that is, it sends sums and products of quaternions to sums and products of matrices. ^[32] In this representation, the conjugate of a quaternion corresponds to the transpose of the matrix. The fourth power of the norm of a quaternion is the determinant of the corresponding matrix. The scalar part of a quaternion is one quarter of the matrix trace. As with the 2 × 2 complex representation above, complex numbers can again be produced by constraining the coefficients suitably; for example, as block diagonal matrices with two 2 × 2 blocks by setting c = d = 0.

Each 4×4 matrix representation of quaternions corresponds to a multiplication table of unit quaternions. For example, the last matrix representation given above corresponds to the multiplication table

which is isomorphic — through ${\displaystyle \{a\mapsto 1,\,b\mapsto i,\,c\mapsto j,\,d\mapsto k\}}$ — to

Constraining any such multiplication table to have the identity in the first row and column and for the signs of the row headers to be opposite to those of the column headers, then there are 3 possible choices for the second column (ignoring sign), 2 possible choices for the third column (ignoring sign), and 1 possible choice for the fourth column (ignoring sign); that makes 6 possibilities. Then, the second column can be chosen to be either positive or negative, the third column can be chosen to be positive or negative, and the fourth column can be chosen to be positive or negative, giving 8 possibilities for the sign. Multiplying the possibilities for the letter positions and for their signs yields 48. Then replacing 1 with a, i with b, j with c, and k with d and removing the row and column headers yields a matrix representation of a + bi + cj + dk.

Lagrange's four-square theorem

Quaternions are also used in one of the proofs of Lagrange's four-square theorem in number theory, which states that every nonnegative integer is the sum of four integer squares. As well as being an elegant theorem in its own right, Lagrange's four square theorem has useful applications in areas of mathematics outside number theory, such as combinatorial design theory. The quaternion-based proof uses Hurwitz quaternions, a subring of the ring of all quaternions for which there is an analog of the Euclidean algorithm.

Quaternions as pairs of complex numbers

Quaternions can be represented as pairs of complex numbers. From this perspective, quaternions are the result of applying the Cayley–Dickson construction to the complex numbers. This is a generalization of the construction of the complex numbers as pairs of real numbers.

Let ${\displaystyle \mathbb {C} ^{2}}$ be a two-dimensional vector space over the complex numbers. Choose a basis consisting of two elements 1 and j. A vector in ${\displaystyle \mathbb {C} ^{2}}$ can be written in terms of the basis elements 1 and j as

$${\displaystyle (a+bi)1+(c+di)\mathbf {j} .}$$

If we define j² = −1 and ij = −ji, then we can multiply two vectors using the distributive law. Using k as an abbreviated notation for the product ij leads to the same rules for multiplication as the usual quaternions. Therefore, the above vector of complex numbers corresponds to the quaternion a + b i + cj + dk. If we write the elements of ${\displaystyle \mathbb {C} ^{2}}$ as ordered pairs and quaternions as quadruples, then the correspondence is

$${\displaystyle (a+bi,\,c+di)\leftrightarrow (a,\,b,\,c,\,d).}$$

Square roots

Functions of a quaternion variable

Like functions of a complex variable, functions of a quaternion variable suggest useful physical models. For example, the original electric and magnetic fields described by Maxwell were functions of a quaternion variable. Examples of other functions include the extension of the Mandelbrot set and Julia sets into 4-dimensional space. ^[36]

Three-dimensional and four-dimensional rotation groups

The word "conjugation", besides the meaning given above, can also mean taking an element a to r a r⁻¹ where r is some nonzero quaternion. All elements that are conjugate to a given element (in this sense of the word conjugate) have the same real part and the same norm of the vector part. (Thus the conjugate in the other sense is one of the conjugates in this sense.) ^[40]

Thus the multiplicative group of nonzero quaternions acts by conjugation on the copy of ${\displaystyle \mathbb {R} ^{3}}$ consisting of quaternions with real part equal to zero. Conjugation by a unit quaternion (a quaternion of absolute value 1) with real part cos(φ) is a rotation by an angle 2φ, the axis of the rotation being the direction of the vector part. The advantages of quaternions are: ^[41]

• Avoiding gimbal lock, a problem with systems such as Euler angles.
• Faster and more compact than matrices.
• Nonsingular representation (compared with Euler angles for example).
• Pairs of unit quaternions represent a rotation in 4D space (see Rotations in 4-dimensional Euclidean space: Algebra of 4D rotations ).

The set of all unit quaternions (versors) forms a 3-sphere S³ and a group (a Lie group) under multiplication, double covering the group ${\displaystyle {\text{SO}}(3,\mathbb {R} )}$ of real orthogonal 3×3  matrices of determinant  1 since two unit quaternions correspond to every rotation under the above correspondence. See plate trick.

The image of a subgroup of versors is a point group, and conversely, the preimage of a point group is a subgroup of versors. The preimage of a finite point group is called by the same name, with the prefix binary. For instance, the preimage of the icosahedral group is the binary icosahedral group.

The versors' group is isomorphic to SU(2), the group of complex unitary 2×2 matrices of determinant 1.

Let A be the set of quaternions of the form a + bi + cj + dk where a, b, c, and d are either all integers or all half-integers. The set A is a ring (in fact a domain) and a lattice and is called the ring of Hurwitz quaternions. There are 24 unit quaternions in this ring, and they are the vertices of a regular 24 cell with Schläfli symbol {3,4,3}. They correspond to the double cover of the rotational symmetry group of the regular tetrahedron. Similarly, the vertices of a regular 600 cell with Schläfli symbol {3,3,5} can be taken as the unit icosians, corresponding to the double cover of the rotational symmetry group of the regular icosahedron. The double cover of the rotational symmetry group of the regular octahedron corresponds to the quaternions that represent the vertices of the disphenoidal 288-cell. ^[42]

Quaternion algebras

The Quaternions can be generalized into further algebras called quaternion algebras. Take F to be any field with characteristic different from 2, and a and b to be elements of F; a four-dimensional unitary associative algebra can be defined over F with basis 1, i, j, and i j, where i² = a, j² = b and i j = −j i (so (i j)² = −a b).

Quaternion algebras are isomorphic to the algebra of 2×2 matrices over F or form division algebras over F, depending on the choice of a and b.

Quaternions as the even part of Cl_3,0(R)

The usefulness of quaternions for geometrical computations can be generalised to other dimensions by identifying the quaternions as the even part ${\displaystyle \operatorname {Cl} _{3,0}^{+}(\mathbb {R} )}$ of the Clifford algebra ${\displaystyle \operatorname {Cl} _{3,0}(\mathbb {R} ).}$ This is an associative multivector algebra built up from fundamental basis elements σ₁, σ₂, σ₃ using the product rules

$${\displaystyle \sigma _{1}^{2}=\sigma _{2}^{2}=\sigma _{3}^{2}=1,}$$$${\displaystyle \sigma _{m}\sigma _{n}=-\sigma _{n}\sigma _{m}\qquad (m\neq n).}$$

If these fundamental basis elements are taken to represent vectors in 3D space, then it turns out that the reflection of a vector r in a plane perpendicular to a unit vector w can be written:

$${\displaystyle r^{\prime }=-w\,r\,w.}$$

Two reflections make a rotation by an angle twice the angle between the two reflection planes, so

$${\displaystyle r^{\prime \prime }=\sigma _{2}\sigma _{1}\,r\,\sigma _{1}\sigma _{2}}$$

corresponds to a rotation of 180° in the plane containing σ₁ and σ₂. This is very similar to the corresponding quaternion formula,

$${\displaystyle r^{\prime \prime }=-\mathbf {k} \,r\,\mathbf {k} .}$$

Indeed, the two structures ${\displaystyle \operatorname {Cl} _{3,0}^{+}(\mathbb {R} )}$ and ${\displaystyle \mathbb {H} }$ are isomorphic. One natural identification is

$${\displaystyle 1\mapsto 1,\quad \mathbf {i} \mapsto -\sigma _{2}\sigma _{3},\quad \mathbf {j} \mapsto -\sigma _{3}\sigma _{1},\quad \mathbf {k} \mapsto -\sigma _{1}\sigma _{2},}$$

and it is straightforward to confirm that this preserves the Hamilton relations

$${\displaystyle \mathbf {i} ^{2}=\mathbf {j} ^{2}=\mathbf {k} ^{2}=\mathbf {i\,j\,k} =-1.}$$

In this picture, so-called "vector quaternions" (that is, pure imaginary quaternions) correspond not to vectors but to bivectors – quantities with magnitudes and orientations associated with particular 2D planes rather than 1D directions. The relation to complex numbers becomes clearer, too: in 2D, with two vector directions σ₁ and σ₂, there is only one bivector basis element σ₁σ₂, so only one imaginary. But in 3D, with three vector directions, there are three bivector basis elements σ₂σ₃, σ₃σ₁, σ₁σ₂, so three imaginaries.

This reasoning extends further. In the Clifford algebra ${\displaystyle \operatorname {Cl} _{4,0}(\mathbb {R} ),}$ there are six bivector basis elements, since with four different basic vector directions, six different pairs and therefore six different linearly independent planes can be defined. Rotations in such spaces using these generalisations of quaternions, called rotors, can be very useful for applications involving homogeneous coordinates. But it is only in 3D that the number of basis bivectors equals the number of basis vectors, and each bivector can be identified as a pseudovector.

There are several advantages for placing quaternions in this wider setting: ^[43]

• Rotors are a natural part of geometric algebra and easily understood as the encoding of a double reflection.
• In geometric algebra, a rotor and the objects it acts on live in the same space. This eliminates the need to change representations and to encode new data structures and methods, which is traditionally required when augmenting linear algebra with quaternions.
• Rotors are universally applicable to any element of the algebra, not just vectors and other quaternions, but also lines, planes, circles, spheres, rays, and so on.
• In the conformal model of Euclidean geometry, rotors allow the encoding of rotation, translation and scaling in a single element of the algebra, universally acting on any element. In particular, this means that rotors can represent rotations around an arbitrary axis, whereas quaternions are limited to an axis through the origin.
• Rotor-encoded transformations make interpolation particularly straightforward.
• Rotors carry over naturally to pseudo-Euclidean spaces, for example, the Minkowski space of special relativity. In such spaces rotors can be used to efficiently represent Lorentz boosts, and to interpret formulas involving the gamma matrices.^{[ citation needed ]}

For further detail about the geometrical uses of Clifford algebras, see Geometric algebra.

Brauer group

The quaternions are "essentially" the only (non-trivial) central simple algebra (CSA) over the real numbers, in the sense that every CSA over the real numbers is Brauer equivalent to either the real numbers or the quaternions. Explicitly, the Brauer group of the real numbers consists of two classes, represented by the real numbers and the quaternions, where the Brauer group is the set of all CSAs, up to equivalence relation of one CSA being a matrix ring over another. By the Artin–Wedderburn theorem (specifically, Wedderburn's part), CSAs are all matrix algebras over a division algebra, and thus the quaternions are the only non-trivial division algebra over the real numbers.

CSAs – finite dimensional rings over a field, which are simple algebras (have no non-trivial 2-sided ideals, just as with fields) whose center is exactly the field – are a noncommutative analog of extension fields, and are more restrictive than general ring extensions. The fact that the quaternions are the only non-trivial CSA over the real numbers (up to equivalence) may be compared with the fact that the complex numbers are the only non-trivial finite field extension of the real numbers.

Quotations

I regard it as an inelegance, or imperfection, in quaternions, or rather in the state to which it has been hitherto unfolded, whenever it becomes or seems to become necessary to have recourse to x, y, z, etc.

—  William Rowan Hamilton (c. 1848) ^[44]

Time is said to have only one dimension, and space to have three dimensions. ... The mathematical quaternion partakes of both these elements; in technical language it may be said to be "time plus space", or "space plus time": And in this sense it has, or at least involves a reference to, four dimensions. ... And how the One of Time, of Space the Three, Might in the Chain of Symbols girdled be.

—  William Rowan Hamilton (c. 1853) ^[45]

Quaternions came from Hamilton after his really good work had been done; and, though beautifully ingenious, have been an unmixed evil to those who have touched them in any way, including Clerk Maxwell.

—  W. Thompson, Lord Kelvin (1892) ^[46]

There was a time, indeed, when I, although recognizing the appropriateness of vector analysis in electromagnetic theory (and in mathematical physics generally), did think it was harder to understand and to work than the Cartesian analysis. But that was before I had thrown off the quaternionic old-man-of-the-sea who fastened himself about my shoulders when reading the only accessible treatise on the subject – Prof. Tait's Quaternions. But I came later to see that, so far as the vector analysis I required was concerned, the quaternion was not only not required, but was a positive evil of no inconsiderable magnitude; and that by its avoidance the establishment of vector analysis was made quite simple and its working also simplified, and that it could be conveniently harmonised with ordinary Cartesian work. There is not a ghost of a quaternion in any of my papers (except in one, for a special purpose). The vector analysis I use may be described either as a convenient and systematic abbreviation of Cartesian analysis; or else, as Quaternions without the quaternions, ... . "Quaternion" was, I think, defined by an American schoolgirl to be "an ancient religious ceremony". This was, however, a complete mistake: The ancients – unlike Prof. Tait – knew not, and did not worship Quaternions.

—  Oliver Heaviside (1893) ^[47]

Neither matrices nor quaternions and ordinary vectors were banished from these ten [additional] chapters. For, in spite of the uncontested power of the modern Tensor Calculus, those older mathematical languages continue, in my opinion, to offer conspicuous advantages in the restricted field of special relativity. Moreover, in science as well as in everyday life, the mastery of more than one language is also precious, as it broadens our views, is conducive to criticism with regard to, and guards against hypostasy [weak-foundation] of, the matter expressed by words or mathematical symbols.

—  Ludwik Silberstein (1924) ^[48]

... quaternions appear to exude an air of nineteenth century decay, as a rather unsuccessful species in the struggle-for-life of mathematical ideas. Mathematicians, admittedly, still keep a warm place in their hearts for the remarkable algebraic properties of quaternions but, alas, such enthusiasm means little to the harder-headed physical scientist.

— Simon L. Altmann (1986) ^[49]

See also

• Conversion between quaternions and Euler angles  – Mathematical strategy
• Dual quaternion  – Eight-dimensional algebra over the real numbers
• Dual-complex number  – Four-dimensional algebra over the real numbersPages displaying short descriptions of redirect targets
• Exterior algebra  – Algebra associated to any vector space
• Hurwitz quaternion order  – Concept in mathematics
• Hyperbolic quaternion  – Mutation of quaternions where unit vectors square to +1
• Lénárt sphere  – Transparent dry-erase sphere used to teach spherical geometry
• Pauli matrices  – Matrices important in quantum mechanics and the study of spin
• Quaternionic manifold  – Concept in geometry
• Quaternionic matrix  – Concept in linear algebra
• Quaternionic polytope  – Concept in geometry
• Quaternionic projective space  – Concept in mathematics
• Rotations in 4-dimensional Euclidean space  – Special orthogonal group
• Slerp  – Spherical linear interpolation in computer graphics
• Split-quaternion  – Four-dimensional associative algebra over the reals
• Tesseract  – Four-dimensional analogue of the cube

Notes

Related Research Articles

In mathematics, the associative property is a property of some binary operations that means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for expressions in logical proofs.

In mathematics, a geometric algebra is an algebra that can represent and manipulate geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the geometric product. Multiplication of vectors results in higher-dimensional objects called multivectors. Compared to other formalisms for manipulating geometric objects, geometric algebra is noteworthy for supporting vector division and addition of objects of different dimensions.

In mathematical physics and mathematics, the Pauli matrices are a set of three 2 × 2 complex matrices that are traceless, Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma, they are occasionally denoted by tau when used in connection with isospin symmetries.

In geometry and physics, spinors are elements of a complex number-based vector space that can be associated with Euclidean space. A spinor transforms linearly when the Euclidean space is subjected to a slight (infinitesimal) rotation, but unlike geometric vectors and tensors, a spinor transforms to its negative when the space rotates through 360°. It takes a rotation of 720° for a spinor to go back to its original state. This property characterizes spinors: spinors can be viewed as the "square roots" of vectors.

In group theory, the quaternion group Q₈ (sometimes just denoted by Q) is a non-abelian group of order eight, isomorphic to the eight-element subset of the quaternions under multiplication. It is given by the group presentation

In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if and are real numbers, then the complex conjugate of is The complex conjugate of is often denoted as or .

In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space under the operation of composition.

Unit quaternions, known as versors, provide a convenient mathematical notation for representing spatial orientations and rotations of elements in three dimensional space. Specifically, they encode information about an axis-angle rotation about an arbitrary axis. Rotation and orientation quaternions have applications in computer graphics, computer vision, robotics, navigation, molecular dynamics, flight dynamics, orbital mechanics of satellites, and crystallographic texture analysis.

In mathematics, the Kronecker product, sometimes denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix. It is a specialization of the tensor product from vectors to matrices and gives the matrix of the tensor product linear map with respect to a standard choice of basis. The Kronecker product is to be distinguished from the usual matrix multiplication, which is an entirely different operation. The Kronecker product is also sometimes called matrix direct product.

In quantum mechanics and computing, the Bloch sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system (qubit), named after the physicist Felix Bloch.

In abstract algebra, the algebra of hyperbolic quaternions is a nonassociative algebra over the real numbers with elements of the form

In abstract algebra, the biquaternions are the numbers w + xi + yj + zk, where w, x, y, and z are complex numbers, or variants thereof, and the elements of {1, i, j, k} multiply as in the quaternion group and commute with their coefficients. There are three types of biquaternions corresponding to complex numbers and the variations thereof:

In mathematics, a versor is a quaternion of norm one. Each versor has the form

In geometry, a three-dimensional space is a mathematical space in which three values (coordinates) are required to determine the position of a point. Most commonly, it is the three-dimensional Euclidean space, that is, the Euclidean space of dimension three, which models physical space. More general three-dimensional spaces are called 3-manifolds. The term may also refer colloquially to a subset of space, a three-dimensional region, a solid figure.

In mathematics, a split-biquaternion is a hypercomplex number of the form

In mathematics, the dual quaternions are an 8-dimensional real algebra isomorphic to the tensor product of the quaternions and the dual numbers. Thus, they may be constructed in the same way as the quaternions, except using dual numbers instead of real numbers as coefficients. A dual quaternion can be represented in the form A + εB, where A and B are ordinary quaternions and ε is the dual unit, which satisfies ε² = 0 and commutes with every element of the algebra. Unlike quaternions, the dual quaternions do not form a division algebra.

In mathematics, the spinor concept as specialised to three dimensions can be treated by means of the traditional notions of dot product and cross product. This is part of the detailed algebraic discussion of the rotation group SO(3).

In mathematical physics, spacetime algebra (STA) is the application of Clifford algebra Cl_1,3(R), or equivalently the geometric algebra G(M⁴) to physics. Spacetime algebra provides a "unified, coordinate-free formulation for all of relativistic physics, including the Dirac equation, Maxwell equation and General Relativity" and "reduces the mathematical divide between classical, quantum and relativistic physics."

In mathematics and mechanics, the Euler–Rodrigues formula describes the rotation of a vector in three dimensions. It is based on Rodrigues' rotation formula, but uses a different parametrization.

In this article, certain applications of the dual quaternion algebra to 2D geometry are discussed. At this present time, the article is focused on a 4-dimensional subalgebra of the dual quaternions which will later be called the planar quaternions.

References

Further reading