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Geometric Matrices
Garret Sobczyk
Universidad de las Am´ericasPuebla
Departamento de F´ısicoMatem´aticas
72820 Puebla, Pue., M´exico
http://www.garretstar.com
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52 Congreso Nacional de La SMM, Oct. 21  25, 2019, Monterrey, NL
1
Complex and Hyperbolic Numbers
Euclidean plane, Hyperbolic plane
z=x+iy, w =x+uy,
where
i2=−1,and u2= 1.
eiθ = cos θ+isin θ, eφu = cosh φ+usinh φ.
eiθ1eiθ2=ei(θ1+θ2), euφ1euφ2=eu(φ1+φ2)
2
Real New Numbers R→R(a,b)
Real numbers R, complex numbers
C, and hyperbolic numbers H.
R−→ Cand R−→ H
Let aand bbe two new numbers
satisfying the Basic Rules
1) a2=0=b2(nilpotents)
2) 2a·b≡ab +ba = 1 (inner
product)
Table 1: Multiplication Table.
a b ab ba
a0ab 0a
b ba 0b0
ab a 0ab 0
ba 0b0ba
aba = (1 −ba)a=a
3
Real numbers Rextended to R(a,b).
X∈R(a,b)⇐⇒
X= ( ba a )x11 x12
x21 x22 ba
b
=x11ba +x21a+x12b+x22ab
We say that [X] = x11 x12
x21 x22 is
the matrix of X.
0.20.40.60.8 1
1
0.5
0.5
1
a
b
e=a+b
f=ab
N
odd
Lorentzean plane R1,1=spanR{e,f}
4
0.20.40.60.8 1
1
0.5
0.5
1
ab
ba
ab+ba=1
abba
N
even
u=2(a b)
Hyperbolic number plane
ab +ba = 1 and ab −ba =u
⇐⇒
(ab)2=ab(1 −ba) = ab
(ba)2=ba(1 −ab) = ba
ab and ba are idempotents.
5
Geometry of R(a,b)
Given
X=(ba a ) [X]ba
b∈R(a,b).
tr(X) := X+X∗=trace[X]
and
det X=XX∗= det[X],
where
X=α+v, X∗=α−v
for α=1
2tr(X)(x11 +x22),
v=x21a+x12b+(x11−x22)b∧a,
and
det X=XX∗= (α+v)(α−v)
=α2−v2=x11x22 −x21x12.
6
Characteristic polynomial of X
ϕ(λ) := (λ−α)2−v2.
CayleyHamilton Theorem
ϕ(X)=(X−α)2−v2= 0.
v2=x12x21 + (x11 −x22)2,
X∈R(a,b) is hyperbolic,parabolic,
or Euclidean if
v2>0
= 0
<0
,
respectively. Null cone v2= 0
1
0.5
0
0.5
1
1 01
1
0
1
1
0.5
0
0.5
w=X−1aX, X−1=X
XX∗
7
Thm i) For X=α+vhyperbolic
X=λ1ˆ
v++λ2ˆ
v−
for real eigenvalues
λ1=α+pv2, λ2=α+pv2,
and eigenpotents (idempotents)
ˆ
v±=1
2(1 ±ˆ
v),ˆ
v≡v
v.
ii) For X=α+nparabolic
X=α+n,where n2= 0.
iii) For X=α+vEuclidean
X=λ1ˆ
v++λ2ˆ
v−
for complex eigenvalues
λ1=α+pv2, λ2=α+pv2,
and eigenpotents (idempotents)
ˆ
v±=1
2(1 ±iˆ
v),ˆ
v≡v
v.
8
Geometric algebras of 2×2 matrices
i) G1,1:= R(e,f)e=M2(R) where
e=a+b,f=a−b
and e2=1=−f2,ef =−fe.
ii) G1,2:= R(e,f, ief)≡C(e,f)e=M2(C)
where
e=a+b,f=a−b
and i=ef(ief) = i(ef)2is a
trivector.
iii) G2:= R(e,ef)e=MR(2) where
e=a+b,f=a−b.
iv) G3:= R(e, if,ef)e=MC(2) where
e=a+b,f=a−b.
Pauli matrices:
[e]=(0 1
1 0 ),[if]=(0−i
i0),[ef]=(1 0
0−1).
9
Spacetime algebra
G3,0=R(e1,e2,e3),
where
e1:= e,e2:= if,e3:= ef.
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Rest frame of an observer (blue)
E:= {e1,e2,e3}
Frame of an observer moving at ve
locity v/c =e2tanh φ(yellow)
E00 := {e00
1,e00
2=e2,e00
3}
10
Stereographic Projection
in G3
x=f(ˆ
a) = 2
ˆ
a+e3
−e3,
e
e
e
1
2
3
m
a
x
e
3

x=x1e1+x2e2∈R2,m=x+e3
Sphere:
S2={ˆ
aˆ
a2= 1} ⊂ R3
11
Stereographic Projection
in G1,2=R(e0,f1,f2)
x=f(ˆ
a) = 2
ˆ
a+e0
−e0,
x=x1f1+x2f2∈R0,2,m=x+e0,
Hyperboloid
L2={ˆ
aˆ
a2= 1} ⊂ R1,2
12
Higher Dimensional Geometric Matrices
Let {a1, b1}and {a2, b2}be pairs
of null vectors satisfying the basic
rules deﬁning the real new num
bers R(a1, b1) and R(a2, b2). In ad
dition, assume that aibk=−bkai
for all i, k = 1,2, anticommuting,
and aiak=−akaiand bibk=−bkai
for i6=k.
Using the Directed Kronecker Product,
the canonical null vector basis
of G2,2over Ris
1
a1−→
⊗1
a2u12 ( 1 b2)←−
⊗( 1 b1)
=
1
a1
a2
a12
u12 ( 1 b1b2b21 )
13
=
u12 b1u2b2u1b21
a1u2u†
1u2a1b2−b2u†
1
a2u1a2b1u1u†
2b1u†
2
a12 −a2u†
1a1u†
2u†
12
.
Given a geometric number g∈
G2,2,
g=( 1 a1a2a12 )u12[g]
1
b1
b2
b21
,
where [g] is the real matrix of g.
14
For g∈G2,3=R(e1, e2, f1, f2, f3),
for ek=ak+bkand fk=ak−bk,
g= ( 1 a1a2a12 )u12[g]
1
b1
b2
b21
,
where [g] is the complex matrix of
g, for i=e1f1e2f2f3.
[a1] = 121
143 =
0 0 0 0
1 0 0 0
0 0 0 0
0 0 1 0
,
[a2] = 131
−142 =
0 0 0 0
0 0 0 0
1 0 0 0
0−1 0 0
,
[b1]=[a1]T,[b2]=[a2]T.
15
More generally, for G5,5e=M32(R)
and G5,6e=M32(C),
[a1] =
12,1
14,3
16,5
18,7
110,9
112,11
114,13
116,15
118,17
120,19
122,21
124,23
126,25
128,27
130,29
132,31
,[a2] =
13,1
−14,2
17,5
−18,6
111,9
−112,10
115,13
−116,14
119,17
−120,18
123,21
−124,22
127,25
−128,26
131,29
−132,30
,
[a3] =
15,1
−16,2
−17,3
18,4
113,9
−114,10
−115,11
116,12
121,17
−122,18
−123,19
124,20
129,25
−130,26
−131,27
132,28
,[a4] =
19,1
−110,2
−111,3
112,4
−113,5
114,6
115,7
−116,8
125,17
−126,18
−127,19
128,20
−129,21
130,22
131,23
−132,24
,
16
[a5] =
117,1
−118,2
−119,3
120,4
−121,5
122,6
123,7
−124,8
−125,9
126,10
127,11
−128,12
129,13
−130,14
−131,15
132,16
,
and [bk]=[ak]Tfor k= 1, ...5.
Gn,n =R(e1,· · · , en, f1,· · · , fn)e=M2n(R)
and
Gn,n+1 =C(e1,· · · , en, f1,· · · , fn)e=M2n(C),
where as before ei:= ai+biand
fi=ai−bi.
17
Bibliography
References
[1] G. Sobczyk, Matrix Gateway to Geometric
Algebra, Spacetime and Spinors, to appear.
[2] ————, Hyperbolic Number Plane, The
College Mathematics Journal, Vol. 26, No. 4,
pp.268280, September 1995.
[3] S. Ramos Ramirez, J.A. Ju´arez Gonz´alez, G.
Sobczyk, From Vectors to Geometric Algebra,
https://arxiv.org/pdf/1802.08153.pdf
[4] G. Sobczyk, New Foundations in Mathe
matics: The Geometric Concept of Number,
Birkh¨auser, New York 2013.
[5] ———, Geometrization of the Real Number
System https://arxiv.org/pdf/1707.02338.pdf
http://www.garretstar.com
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