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Prolog

What is the square root of -1?

Hypercomplex Numbers

ak's are real numbers, ik are different complex units with the property (ik)2 = -1 for all k.

n Complex Units Name Structure
0 0 Real Numbers a0
1 1 Complex Numbers a0 + a1 i1
2 3 Quaternions a0 + a1 i1 + a2 i2 + a3 i3
3 7 Octonions a0 + a1 i1 + a2 i2 + a3 i3 + a4 i4 + a5 i5 + a6 i6+ a7 i7
4 15 Sedenions a0 + a1 i1 + a2 i2 + a3 i3 + a4 i4 + a5 i5 + a6 i6+ a7 i7 + a8 i8 + a9 i9 + a10 i10 + a11 i11 + a12 i12 + a13 i13 + a14 i14 + a15 i15
n 2n-1 a0 + ... + an2-1 in2-1

Basic Matrices and Complex Numbers

The following 2x2 matrices can be used to construct a matrix representation of the complex numbers and a matrix representation of the quaternions.

With the basic properties: U2 = I; H2 = I; B2 = -I

These are the same Matrices that where used for reflection and 90 degree rotation on the rotation matrix page.

Matrix Representation of Complex Numbers

The matrix equation z = xI + yB is isomorphic to the complex numbers. For more Information on these matrices see chromogeometry on my Geometric Algebra and Matrices page. For information of the realation between geometric algebra and complex numbers see my geometric algebra page.

With these matrices it is also possible to represent the geometric algebra 𝔾2 in matrix form. With the base vectors as matrices: e1 = U; e2 = H; e12 = B.

These Matrices are also used to perform basic reflections and 90° rotations in the plane.

Quaternions

Quaternions are elements of the Form:

Quaternions

q0, q1, q2, q3 are real numbers. i, j, k are three different square roots of -1 and are the new elements used for the construction of quaternions. They have the following algebraic properties:

Algebraic Properties

Quaternion Basic

From this fundamental relations we get:

i equations

j equations

k equations

From this relations a multiplication table can be constructed.

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

Complex Conjugate of a Quaternion

Quaternion Conjugated

Square of the Length

Length Square

The Product of two Quaternions

P-Quaternion

Q-Quaternion

Quaternions Product

The Relation between Quaternions and Geometic Algebra

The basic elements i, j, k of the quarterions can be indentified with the following bivectors in geometric algebra.

i quaternion

j quaternion

k quaternion

The elements i, j, k form a left handed System.

i, j, k, product

three bivectors left handed

On the other hand if I build the product of bivectors for a right handed system I get:

three bivectors right handed

If we build the geometric product of two vectors u and v from R^2

Geometric Product Canonic

Will result in the following quarternion:

Geometric Product Canonic

Confusion in the Application of Quaternions

Hamilton tried to apply i, j, k as Vectors in a right handed System, which lead to much confusion in such applications. This confusion can be avoided if the the i, j, k are interpretated as bivectors in a left handed system.

In older books sometimes i, j. k can be found to name the three spatial unit vectors in a right handed system. Which is correct, if one does not interpret them as elements of a quaternion. Hamilton introduced vectors into the main stream of mathematics in this form. Gibbs developed the vector analysis to overcome the difficulties introduced by the misinterpretation of quaternions. On the other hand quaternions are an powerfull tool to perform rotations.

Biquaternions

Quaternions and Biquaternions are elements of the Form:

Quaternions

For quaternions   q0, q1, q2, q3 are real numbers.

For biquaternions q0, q1, q2, q3 are complex numbers.

Quaternions in Matrix Form

Quaternions as 2x2 matrices

The presentation of quaternions as 2x2 matrices with complex matrix elements looks as follows:

Quaternions as 4x4 Matrices

Replaching the elements of the 2x2 quaternion matrices by the following 2x2 matrices...

...will result in the following 4x4 matrices:

These matrices have the same properties as the quaternions:

i2 = j2 = k2 = ijk = -I

ij =  k    jk =  i    ki =  j   
ji = -k    kj = -i    ik = -j   

These relations will result in the following matrix representation by 4x4 matrices:

The Presentation of the quaternions by 4x4 matrices is not unique. Exchanging the elements in the Kronecker product will give another matrix representation of the quaternions:

Base Matrices of the Quaternions

The quaternionen base elements represented as 4x4 matrices can generated by the Kronecker product of the basic 2x2 matrices I, U, H and B. The presentation of the quaternion base elements is not unique. There are several possibilities to implement quaternions by 4x4 matrices.

Example 1

Example 2

Matrix Presentation of a Quaternion

q = aI + bi + cj + dk

The presentation of a quaternion in matrix form is not unique.

Example 1

Example 2

Pauli Matrices

The Pauli matrices contain the complex element i with the property i2 = -1.

More on Pauli matrices as 2x2 matrices with complex elements can be found here:

Pauli Matrices as 4x4 Matrices

Replaching the elements i, 1, 0 of the Pauli matrices by the following 2x2 matrices:

Will result into the following 4x4 matrices:

Matrix Representation of the Canonic Base Elements of the Geometric Algebra 𝔾3

Geometric algebra 𝔾3 by 4x4 matrices. The representation of the Geometric Algebra 𝔾3 by 4x4 matrices is not unique.

Example 1

Example 2

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Kronecker Products

Kronecker products of the base matrices I, U, H and B.

The Cayley-Dickson Construction

A complex number z = a+bi can be represented as an ordered pair (a,b) of real numbers. The product of two complex numbers z1 = (a,b) and z2 = (c,d) is then defined as follows:

z1z2 = (a,b)(c,d)=(ac-bd, ad+cb)

The complex conjugate of a complex number z is defined as follows:

z = (a,b) = (a,-b)

Using the matrix representation of two complex numbers z1, z2 will give the following product:

The elements a,b,c,d are real numbers.

Using complex numbers instead of real numbers will result in the construction of the quaternions.

q = (z1, z2)

In this case the following, modified product has to be applied:

q1q2 = (z1, z2)(z3, z4) = (z1z3-z2z4, z1z4+z3z2) = q3

This product will have the following matrix representation. In a first step the product of the two matrices with complex components is executed. Then all the required conjugations are executed.

The 2x2 matrices with complex components can be written as 4x4 matrices with real components, by using the substitution:

General Construction of Hypercomples Numbers

If a set of hypercomplex numbers is given the next set of hypercomplex numbers can be construted by using an order pair of the given hyper complex numbers (a,b) to construct the next higher set of hypercomplex numbers by applying the following rules:

(a,b)(c,d)= (ac-bd, ad+cb)

With the complex conjungate for hypercomplex numbers is defined as follows:

(a,b) = (a,-b)

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Octonions

1

o1

o2

o3

o4

o5

o6

o7

1

1

o1

o2

o3

o4

o5

o6

o7

o1

o1

-1

o3

-o2

o5

-o4

-o7

o6

o2

o2

-o3

-1

o1

o6

o7

-o4

-o5

o3

o3

o2

-o1

-1

o7

-o6

o5

-o4

o4

o4

-o5

-o6

-o7

-1

o1

o2

o3

o5

o5

o4

-o7

o6

-o1

-1

-o3

o2

o6

o6

o7

o4

-o5

-o2

o3

-1

-o1

o7

o7

-o6

o5

o4

-o3

-o2

o1

-1

Sedenions

1 s1 s2 s3 s4 s5 s6 s7 s8 s9 s10 s11 s12 s13 s14 s15
1 1 s1 s2 s3 s4 s5 s6 s7 s8 s9 s10 s11 s12 s13 s14 s15
s1 s1 -1 s3 -s2 s5 -s4 -s7 s6 s9 -s8 -s11 s10 -s13 s12 s15 -s14
s2 s2 -s3 -1 s1 s6 s7 -s4 -s5 s10 s11 -s8 -s9 -s14 -s15 s12 s13
s3 s3 s2 -s1 -1 s7 -s6 s5 -s4 s11 -s10 s9 -s8 -s15 s14 -s13 s12
s4 s4 -s5 -s6 -s7 -1 s1 s2 s3 s12 s13 s14 s15 -s8 -s9 -s10 -s11
s5 s5 s4 -s7 s6 -s1 -1 -s3 s2 s13 -s12 s15 -s14 s9 -s8 s11 -s10
s6 s6 s7 s4 -s5 -s2 s3 -1 -s1 s14 -s15 -s12 s13 s10 -s11 -s8 s9
s7 s7 -s6 s5 s4 -s3 -s2 s1 -1 s15 s14 -s13 -s12 s11 s10 -s9 -s8
s8 s8 -s9 -s10 -s11 -s12 -s13 -s14 -s15 -1 s1 s2 s3 s4 s5 s6 s7
s9 s9 s8 -s11 s10 -s13 s12 s15 -s14 -s1 -1 -s3 s2 -s5 s4 s7 -s6
s10 s10 s11 s8 -s9 -s14 -s15 s12 s13 -s2 s3 -1 -s1 -s6 -s7 s4 s5
s11 s11 -s10 s9 s8 -s15 s14 -s13 s12 -s3 -s2 s1 -1 -s7 s6 -s5 s4
s12 s12 s13 s14 s15 s8 -s9 -s10 -s11 -s4 s5 s6 s7 -1 -s1 -s2 -s3
s13 s13 -s12 s15 -s14 s9 s8 s11 -s10 -s5 -s4 s7 -s6 s1 -1 s3 -s2
s14 s14 -s15 -s12 s13 s10 -s11 s8 s9 -s6 -s7 -s4 s5 s2 -s3 -1 s1
s15 s15 s14 -s13 -s12 s11 s10 -s9 s8 -s7 s6 -s5 -s4 s3 s2 -s1 -1
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The next page is about Finite Groups.

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8. January 2021 Version 2.0
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