times, .*

Element-wise quaternion multiplication

Description

example

quatC = A.*B returns the element-by-element quaternion multiplication of quaternion arrays.

You can use quaternion multiplication to compose rotation operators:

• To compose a sequence of frame rotations, multiply the quaternions in the same order as the desired sequence of rotations. For example, to apply a p quaternion followed by a q quaternion, multiply in the order pq. The rotation operator becomes ${\left(pq\right)}^{\ast }v\left(pq\right)$, where v represents the object to rotate in quaternion form. * represents conjugation.

• To compose a sequence of point rotations, multiply the quaternions in the reverse order of the desired sequence of rotations. For example, to apply a p quaternion followed by a q quaternion, multiply in the reverse order, qp. The rotation operator becomes $\left(qp\right)v{\left(qp\right)}^{\ast }$.

Examples

collapse all

Create two vectors, A and B, and multiply them element by element.

A = quaternion([1:4;5:8]);
B = A;
C = A.*B
C = 2x1 quaternion array
-28 +   4i +   6j +   8k
-124 +  60i +  70j +  80k

Create two 3-by-3 arrays, A and B, and multiply them element by element.

A = reshape(quaternion(randn(9,4)),3,3);
B = reshape(quaternion(randn(9,4)),3,3);
C = A.*B
C = 3x3 quaternion array
0.60169 +  2.4332i -  2.5844j + 0.51646k    -0.49513 +  1.1722i +  4.4401j -   1.217k      2.3126 + 0.16856i +  1.0474j -  1.0921k
-4.2329 +  2.4547i +  3.7768j + 0.77484k    -0.65232 - 0.43112i -  1.4645j - 0.90073k     -1.8897 - 0.99593i +  3.8331j + 0.12013k
-4.4159 +  2.1926i +  1.9037j -  4.0303k     -2.0232 +  0.4205i - 0.17288j +  3.8529k     -2.9137 -  5.5239i -  1.3676j +  3.0654k

Note that quaternion multiplication is not commutative:

isequal(C,B.*A)
ans = logical
0

Create a row vector a and a column vector b, then multiply them. The 1-by-3 row vector and 4-by-1 column vector combine to produce a 4-by-3 matrix with all combinations of elements multiplied.

a = [zeros('quaternion'),ones('quaternion'),quaternion(randn(1,4))]
a = 1x3 quaternion array
0 +       0i +       0j +       0k           1 +       0i +       0j +       0k     0.53767 +  1.8339i -  2.2588j + 0.86217k

b = quaternion(randn(4,4))
b = 4x1 quaternion array
0.31877 +   3.5784i +   0.7254j -  0.12414k
-1.3077 +   2.7694i - 0.063055j +   1.4897k
-0.43359 -   1.3499i +  0.71474j +    1.409k
0.34262 +   3.0349i -  0.20497j +   1.4172k

a.*b
ans = 4x3 quaternion array
0 +        0i +        0j +        0k      0.31877 +   3.5784i +   0.7254j -  0.12414k      -4.6454 +   2.1636i +   2.9828j +   9.6214k
0 +        0i +        0j +        0k      -1.3077 +   2.7694i - 0.063055j +   1.4897k      -7.2087 -   4.2197i +   2.5758j +   5.8136k
0 +        0i +        0j +        0k     -0.43359 -   1.3499i +  0.71474j +    1.409k       2.6421 -     5.32i -   2.3841j -   1.3547k
0 +        0i +        0j +        0k      0.34262 +   3.0349i -  0.20497j +   1.4172k      -7.0663 -  0.76439i -  0.86648j +   7.5369k

Input Arguments

collapse all

Array to multiply, specified as a quaternion, an array of quaternions, a real scalar, or an array of real numbers.

A and B must have compatible sizes. In the simplest cases, they can be the same size or one can be a scalar. Two inputs have compatible sizes if, for every dimension, the dimension sizes of the inputs are the same or one of them is 1.

Data Types: quaternion | single | double

Array to multiply, specified as a quaternion, an array of quaternions, a real scalar, or an array of real numbers.

A and B must have compatible sizes. In the simplest cases, they can be the same size or one can be a scalar. Two inputs have compatible sizes if, for every dimension, the dimension sizes of the inputs are the same or one of them is 1.

Data Types: quaternion | single | double

Output Arguments

collapse all

Quaternion product, returned as a scalar, vector, matrix, or multidimensional array.

Data Types: quaternion

Algorithms

collapse all

Quaternion Multiplication by a Real Scalar

Given a quaternion,

$q={a}_{\text{q}}+{b}_{\text{q}}\text{i}+{c}_{\text{q}}\text{j}+{d}_{\text{q}}\text{k,}$

the product of q and a real scalar β is

$\beta q=\beta {a}_{\text{q}}+\beta {b}_{\text{q}}\text{i}+\beta {c}_{\text{q}}\text{j}+\beta {d}_{\text{q}}\text{k}$

Quaternion Multiplication by a Quaternion Scalar

The definition of the basis elements for quaternions,

${\text{i}}^{2}={\text{j}}^{2}={\text{k}}^{2}=\text{ijk}=-1\text{\hspace{0.17em}},$

can be expanded to populate a table summarizing quaternion basis element multiplication:

 1 i j k 1 1 i j k i i −1 k −j j j −k −1 i k k j −i −1

When reading the table, the rows are read first, for example: ij = k and ji = −k.

Given two quaternions, $q={a}_{\text{q}}+{b}_{\text{q}}\text{i}+{c}_{\text{q}}\text{j}+{d}_{\text{q}}\text{k,}$ and $p={a}_{\text{p}}+{b}_{\text{p}}\text{i}+{c}_{\text{p}}\text{j}+{d}_{\text{p}}\text{k}$, the multiplication can be expanded as:

$\begin{array}{l}z=pq=\left({a}_{\text{p}}+{b}_{\text{p}}\text{i}+{c}_{\text{p}}\text{j}+{d}_{\text{p}}\text{k}\right)\left({a}_{\text{q}}+{b}_{\text{q}}\text{i}+{c}_{\text{q}}\text{j}+{d}_{\text{q}}\text{k}\right)\\ \text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}={a}_{\text{p}}{a}_{\text{q}}+{a}_{\text{p}}{b}_{\text{q}}\text{i}+{a}_{\text{p}}{c}_{\text{q}}\text{j}+{a}_{\text{p}}{d}_{\text{q}}\text{k}\\ \text{ }\text{ }+{b}_{\text{p}}{a}_{\text{q}}\text{i}+{b}_{\text{p}}{b}_{\text{q}}{\text{i}}^{2}+{b}_{\text{p}}{c}_{\text{q}}\text{ij}+{b}_{\text{p}}{d}_{\text{q}}\text{ik}\\ \text{ }\text{ }+{c}_{\text{p}}{a}_{\text{q}}\text{j}+{c}_{\text{p}}{b}_{\text{q}}\text{ji}+{c}_{\text{p}}{c}_{\text{q}}{\text{j}}^{2}+{c}_{\text{p}}{d}_{\text{q}}\text{jk}\\ \text{ }\text{ }+{d}_{\text{p}}{a}_{\text{q}}k+{d}_{\text{p}}{b}_{\text{q}}\text{ki}+{d}_{\text{p}}{c}_{\text{q}}\text{kj}+{d}_{\text{p}}{d}_{\text{q}}{\text{k}}^{2}\end{array}$

You can simplify the equation using the quaternion multiplication table.

$\begin{array}{l}z=pq\text{\hspace{0.17em}}={a}_{\text{p}}{a}_{\text{q}}+{a}_{\text{p}}{b}_{\text{q}}\text{i}+{a}_{\text{p}}{c}_{\text{q}}\text{j}+{a}_{\text{p}}{d}_{\text{q}}\text{k}\\ \text{ }\text{ }+{b}_{\text{p}}{a}_{\text{q}}\text{i}-{b}_{\text{p}}{b}_{\text{q}}+{b}_{\text{p}}{c}_{\text{q}}\text{k}-{b}_{\text{p}}{d}_{\text{q}}\text{j}\\ \text{ }\text{ }+{c}_{\text{p}}{a}_{\text{q}}\text{j}-{c}_{\text{p}}{b}_{\text{q}}\text{k}-{c}_{\text{p}}{c}_{\text{q}}+{c}_{\text{p}}{d}_{\text{q}}\text{i}\\ \text{ }\text{ }+{d}_{\text{p}}{a}_{\text{q}}k+{d}_{\text{p}}{b}_{\text{q}}\text{j}-{d}_{\text{p}}{c}_{\text{q}}\text{i}-{d}_{\text{p}}{d}_{\text{q}}\end{array}$

 Kuipers, Jack B. Quaternions and Rotation Sequences: A Primer with Applications to Orbits, Aerospace, and Virtual Reality. Princeton, NJ: Princeton University Press, 2007.