Maths - Pauli matrices - 2D

The basis vectors can be represented by matrices, this algebra was worked out independently by Pauli for his work on quantum mechanics, we can define the following equivalents:

e1=
0 1
1 0
e2=
1 0
0 -1

The scalar would be the identity matrix.

1=
1 0
0 1

The bivector can be calculated by multiplying the matrices:

e1e2=
0 1
1 0
*
1 0
0 -1
=
0 -1
1 0

So the complete geometric multiplication table is:

a*b
b.e b.e1 b.e2 b.e12
a.e
1 0
0 1
0 1
1 0
1 0
0 -1
0 -1
1 0
a.e1
0 1
1 0
1 0
0 1
0 -1
1 0
1 0
0 -1
a.e2
1 0
0 -1
0 1
-1 0
1 0
0 1
0 -1
-1 0
a.e12
0 -1
1 0
-1 0
0 1
0 1
1 0
-1 0
0 -1

which is equivilant to the table derived here.

 

Any multivector can be represented by a single matrix representing a linear sum of these basis, for instance,

4 3
3 -4
= 3*
0 1
1 0
+ 4*
1 0
0 -1
= 3e1+ 4e2

Further Reading

Other uses of Pauli Matrix:

Related Concepts:


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see also:

 

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