Wacker Art Geometric Algebra and Matrices Wappen der Familie Wacker
Passat Travemünde
Bild: "Passat Travemünde"

Prolog

This page is based on the concepts developed on my geometric algebra page and on my linear algebra page.

Determinants and the Outer Product

Outer Product of two Vectors and the Determinant

Bulding the outer product of two vectors u and v gives:

uv = (u1e1 + u2e2) ∧ (v1e1 + v2e2) = (u1v2 - u2v1)(e1e2);

The first term on the right side (u1v2 - u2v1) is the value of the determinant:

Outer Product of three Vectors and the Determinant

Bulding the outer product of three vectors u, v and w gives:

uvw = (u1e1 + u2e2 + u3e3) ∧ (v1e1 + v2e2 + v3e3) ∧ (w1e1 + w2e2 + w3e3) = (u1v2w3 + v1w2u3 + w1u2v3 - w1v2u3 - v1u2w3 - u1w2v3) (e1e2e3);

The first term on the right side (u1v2w3 + v1w2u3 + w1u2v3 - w1v2u3 - v1u2w3 - u1w2v3) is the value of the determinant.

Outer Product of n-Vectors and the Determinant

The outer product of n-vectors is the derminant of the matrix with the n vectors as column vectors multiplied with the unit pseudoscalar I(n)

Gateway to Paradise
Bild: "Gateway to Paradise"

Complex and Hyperbolic Numbers in Matrix Form

Complex Numbers as Matrices

A complex unit called i is the solution of the expression:

x2 + 1 = 0

and has the property:

i2 = -1

A complex number can be written as a matrix.

The complex conjugate in matrix form:

Complex Units in Matrixs Form

Hyperbolic Numbers as Matrices

Hyperbolic unit called u is the solution of the expression:

x2 - 1 = 0

u2 = 1 and u ≠ ±1

A hyperbolic number can be expressed as matrix.

Hyperbolic Units in Matrix Form

u(-u) = (-u)u = -I

Geometric Numbers

Geometric numbers comprise the real numbers and two new elements a,b with special properties. The new geometric numbers a,b also called g-numbers have the following properties:

(i)  a2=0=b2 The new g-numbers are called nilpotent or null vectors.

(ii) 2ab =ab+ba=1

The g-Numbers Expressed as Matrices

The canonic elements a and b expressed as matrices:

Squaring of the canonic matrices will result in the zero matrix:


a2 = 0;

b2 = 0;

Building the product a and b and the product b and a.

ab

ba

The products ab and ba expressed as matrices:

ab + ba = 1

ab - ba = u

aba=(ab)a=(1-ba)a=a-ba2=a

bab=(ba)b=(1-ab)b=b-ab2=b

(ab)2=(1-ba)ab=ab+ba2b=ab

(ba)2=(1-ab)ba=ba+ab2a=ba

(ab)(ba) = 0 = (ba)(ab)

From the above properties the multiplication table for basic g-numbers a, b, ab, ba can be dereived.

a b ab ba
a 0 ab 0 a
b ba 0 b 0
ab a 0 ab 0
ba 0 b 0 ba

The basic g-numbers can be writen in a matrix form:

We have the following matrices for a, b, ab and ba:

The general form of a g-matrix is as follows:

A geometric number g can know be expressed as follows.

Conjugation and Inversion

Each g-number can be seen as the sum of two parts g = go + ge the odd part go and the even part ge.

The odd part of a g-number looks as follows:

The even part of a g-number looks as follows:

The reverse of a g-number

With (ab) = ba the reverse operation for the odd and the even part are defined as follows:

go = go

ge = g11ab + g22ba

The reverse of a g-number has the following property:

g = (go + ge) = go + ge = go + ge

Hence we have for g

The inverse of a g-number

The invers of a g-numger is named g- and has the following properties:

g-o = -g12b - g21a; for the odd part.

g-e = ge; for the even part.

The sum of the even and the odd part will give g-

g- = (go + ge)- = g-o + g-e = -go + ge

Hence we have for g-

Mixed conjugation

The combination of the operations of reversion and inversion of a g-number will give the mixed conjugation g*.

g* = (g)- = (go + ge)- = -go + ge

Ship Flensburg
Picture: "Old Ship Flensburg"
Counter
18. Dezember 2017 Version 1.0
Copyright: Hermann Wacker Uhlandstraße 10 D-85386 Eching bei Freising Germany Haftungsausschluß