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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 q = a + b i + c j + d k {\displaystyle q=a+b\,\mathbf {i} +c\,\mathbf {j} +d\,\mathbf {k} } be a quaternion. The conjugate of q is the quaternion q ∗ = a − b i − c j − d k {\displaystyle q^{*}=a-b\,\mathbf {i} -c\,\mathbf {j} -d\,\mathbf {k} }. It is denoted by q, q, q ~ {\displaystyle {\tilde {q}}}, or q. 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:

q ∗ = − 1 2 (q + i q i + j q j + k q k). {\displaystyle q^{*}=-{\frac {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:

‖ q ‖ = q q ∗ = q ∗ q = a 2 + b 2 + c 2 + d 2 {\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 H {\displaystyle \mathbb {H} } considered as the vector space R 4 {\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

‖ α q ‖ = | α | ‖ q ‖. {\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

‖ p q ‖ = ‖ p ‖ ‖ q ‖ {\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

det (a + i b i d + c i d − c a − i b) = a 2 + b 2 + c 2 + d 2, {\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 distance d (p, q) between p and q as the norm of their difference:

d (p, q) = ‖ p − q ‖. {\displaystyle d(p,q)=\lVert p-q\rVert ~.}

This makes H {\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 R {\displaystyle \mathbb {R} } from the fact that H {\displaystyle \mathbb {H} } is a normed algebra.