Prolog
This page is based on the concepts developed on my geometric algebra page and on my linear algebra page.
Home | Gallery | Physik | Philosophie | Software | Musik | Kunst | H.Wacker | Links | Inhalt | English |
Mathematik | Geometric Algebra | Clifford Algebra | Lineare Algebra | Primzahlen |
This page is based on the concepts developed on my geometric algebra page and on my linear algebra page.
Bulding the outer product of two vectors u and v gives:
u ∧ v = (u_{1}e_{1} + u_{2}e_{2}) ∧ (v_{1}e_{1} + v_{2}e_{2}) = (u_{1}v_{2} - u_{2}v_{1})(e_{1}∧e_{2});
The first term on the right side (u_{1}v_{2} - u_{2}v_{1}) is the value of the determinant:
Bulding the outer product of three vectors u, v and w gives:
u ∧ v ∧ w = (u_{1}e_{1} + u_{2}e_{2} + u_{3}e_{3}) ∧ (v_{1}e_{1} + v_{2}e_{2} + v_{3}e_{3}) ∧ (w_{1}e_{1} + w_{2}e_{2} + w_{3}e_{3}) = (u_{1}v_{2}w_{3} + v_{1}w_{2}u_{3} + w_{1}u_{2}v_{3} - w_{1}v_{2}u_{3} - v_{1}u_{2}w_{3} - u_{1}w_{2}v_{3}) (e_{1}∧e_{2}∧e_{3});
The first term on the right side (u_{1}v_{2}w_{3} + v_{1}w_{2}u_{3} + w_{1}u_{2}v_{3} - w_{1}v_{2}u_{3} - v_{1}u_{2}w_{3} - u_{1}w_{2}v_{3}) is the value of 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)}
A complex unit called i is the solution of the expression:
x^{2} + 1 = 0
and has the property:
i^{2} = -1
A complex number can be written as a matrix.
The complex conjugate in matrix form:
Hyperbolic unit called u is the solution of the expression:
x^{2} - 1 = 0
u^{2} = 1 and u ≠ ±1
A hyperbolic number can be expressed as matrix.
u(-u) = (-u)u = -I
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) a^{2}=0=b^{2} The new g-numbers are called nilpotent or null vectors.
(ii) 2a‧b =ab+ba=1
The canonic elements a and b expressed as matrices:
Squaring of the canonic matrices will result in the zero matrix:
a^{2} = 0;
b^{2} = 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-ba^{2}=a
bab=(ba)b=(1-ab)b=b-ab^{2}=b
(ab)^{2}=(1-ba)ab=ab+ba^{2}b=ab
(ba)^{2}=(1-ab)ba=ba+ab^{2}a=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.
Each g-number can be seen as the sum of two parts g = g_{o} + g_{e} the odd part g_{o} and the even part g_{e}.
The odd part of a g-number looks as follows:
The even part of a g-number looks as follows:
With (ab)^{†} = ba the reverse operation for the odd and the even part are defined as follows:
g_{o}^{†} = g_{o}
g_{e}^{†} = g_{11}ab + g_{22}ba
The reverse of a g-number has the following property:
g^{†} = (g_{o} + g_{e})^{†} = g_{o}^{†} + g_{e}^{†} = g_{o} + g_{e}^{†}
Hence we have for g^{†}
The invers of a g-numger is named g^{-} and has the following properties:
g^{-}_{o} = -g_{12}b - g_{21}a; for the odd part.
g^{-}_{e} = g_{e}; for the even part.
The sum of the even and the odd part will give g^{-}
g^{-} = (g_{o} + g_{e})^{-} = g^{-}_{o} + g^{-}_{e} = -g_{o} + g_{e}
Hence we have for g^{-}
The combination of the operations of reversion and inversion of a g-number will give the mixed conjugation g^{*}.
g^{*} = (g^{†})^{-} = (g_{o} + g_{e}^{†})^{-} = -g_{o} + g_{e}^{†}
Home | Gallery | Physik | Philosophie | Software | Musik | Kunst | H.Wacker | Links | Inhalt | English |
Mathematik | Geometric Algebra | Clifford Algebra | Linear Algebra | Primzahlen |