Dual
This multiplies by e12 , for general information about n-dimentional dual see this page.
Ar |
Ar* = e12 Ar |
1 | e12 |
e1 | e12e1= -e2 |
e2 | e12e2= e1 |
e12 | e12e12= -1 |
Reverse
This reverses the order of the indexes, for general information about n-dimentional reverse see this page.
Ar |
Ar† |
1 | 1 |
e1 | e1 |
e2 | e2 |
e12 | e21 = -e12 |
Conjugate
The conjugate of Ar is denoted Ar~ where: Ar~*Ar = I = psudoscalar, for general information about n-dimentional conjugate see this page.
If Ar represents a transformation then Ar~ reverses the transformation
Ar |
Ar~=(Ar†)* | Ar~*Ar |
1 | (1)*= e12 | e12 |
e1 | (e1)*= -e2 | e12 |
e2 | (e2)*= e1 | e12 |
e12 | (-e12)*= 1 | e12 |
The last column confirms that Ar~*Ar = I = psudoscalar.
Inverse
try multipy by conjugate:
(e + ex + ey + exy)( e + ex + ey - exy)
a.e | a.ex | a.ey | -a.exy | |
b.e | e | ex | ey | -exy |
b.ex | ex | e | exy | -ey |
b.ey | ey | -exy | e | +ex |
b.exy | exy | -ey | ex | e |
e = 4e
ex = 4ex
ey = 0
exy = -exy+exy-exy+exy =0
a.e | a.ex | a.ey | a.exy | |
b.e | e | ex | ey | exy |
b.ex | ex | e | exy | ey |
b.ey | ey | -exy | e | -ex |
b.exy | exy | -ey | ex | -e |