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times, .*

Element-wise quaternion multiplication

Syntax

quatC = A.*B

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 (pq)v(pq), 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 (qp)v(qp).

Examples

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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

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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

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

Data Types: quaternion

Algorithms

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Quaternion Multiplication by a Real Scalar

Given a quaternion,

q=aq+bqi+cqj+dqk,

the product of q and a real scalar β is

βq=βaq+βbqi+βcqj+βdqk

Quaternion Multiplication by a Quaternion Scalar

The definition of the basis elements for quaternions,

i2=j2=k2=ijk=-1,

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

 1ijk
11ijk
ii−1k−j
jj−k−1i
kkj−i−1

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

Given two quaternions, q=aq+bqi+cqj+dqk, and p=ap+bpi+cpj+dpk, the multiplication can be expanded as:

z=pq=(ap+bpi+cpj+dpk)(aq+bqi+cqj+dqk)=apaq+apbqi+apcqj+apdqk+bpaqi+bpbqi2+bpcqij+bpdqik+cpaqj+cpbqji+cpcqj2+cpdqjk+dpaqk+dpbqki+dpcqkj+dpdqk2

You can simplify the equation using the quaternion multiplication table.

z=pq=apaq+apbqi+apcqj+apdqk+bpaqibpbq+bpcqkbpdqj+cpaqjcpbqkcpcq+cpdqi+dpaqk+dpbqjdpcqidpdq

References

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

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

Introduced in R2018a