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Prolog

Space, the final frontier.

Introduction

Every finit group is a set of permutations. The fact, that every finit group is a set of permutations is called Cayley's theorem. It is named after Arthur Cayley, who presented this fact to the world in 1854. "On the theory of groups, as depending on the symbolic equation θⁿ = 1" - Philosophical Magazine. Cayley sees the elements of groups as n^th root of unity.

Permutation Matrices

A permutation can be represented by a matrix operation. A permutation matrix is a square matrix with a single one in each row and a single one in each column and zeros everywhere else. Multiplying a permutation matrix with a vector, will result in a vector, where only the order of the vector elements has changed.

Example

Two distinguished elements {a,b} can have only two permutations: (a,b) and (b,a)

Multiplying the 2x2 exchange matrix with the Vector (a,b)^T will result in the vector (b,a)^T.

Definition of a Group

The definition of a group can be found on the abstract algebra page.

Abstract Algebra

Cyclic Group C²

There are many representations of groups by 2x2 matrices on the geometric algebra and matrices page. The matrices for reflection an rotation can be found on the rotation matrix page. The matrices for 90° rotation and reflection around the x-axis and the y=x line are basic building elements for many other matrices.

Rotation Matrices	Hyperbolic Numbers	Complex Numbers	Geometric Numbers	Chromogeometry
Rotation and reflections in the plane can be expressed by 2x2 matrices.	Presentation of hyperbolic numbers by 2x2 matrices.	Presentation of complex numbers by 2x2 matrices.	Geometric Numbers are represented by matrices of the following form:	Chromogeometrie contains the matrices of hyperbolic numbers, complex numbers and a new kind of hyperbolic matrices.
Rotation Matrices	Hyperbolic Numbers	Complex Numbers	Geometric Numbers	Chromogeometry

Commutator and Anticommutator

The matrix product of two different matrices A and B is in general noncommutative: AB ≠ BA.

Commutator

The commutator brackes of two matrices are defined as follows:

[A,B] = AB - BA

If AB ≠ BA then AB - BA ≠ 0.

Anticommutator

The anticommutator brackets of two matrices are defined as follows:

{A,B} = AB + BA

If AB ≠ BA then AB + BA ≠ 2AB.

Examples:

Let A and B be 2x2 matrices with the following values:

The Exponential of a Matrix

Exponential Series

The matrix exponential e^A means that the matrix is in the exponent.
The matrix exponential is defined by a inifinit series of matrix products, analog to the series for real numbers:

Bold face notation is used to the exponential of a matrix. When A is a nxn matrix then e^A is a nxn matrix.

Commuative Matrices

A and B are matrices with AB = BA (the matrices are commutative) then

e^A+B = e^Ae^B

Invers of a Exponential Matrix

e^-Ae^A = e^Ae^-A = e^A-A = e⁰ = I;

Diagonal Matrices

If a matrix has only elements on the Diagonal we have:

Invertibal Matrices

If P is an invertibal nxn matrix and A is a nxn matrix we have:

Pe^AP^-1 = e^PAP-1

Trace of the Exponential Determinant

For quadratic matrices A we have:

det(e^A) = e^trace(A)

This leads to the special case when the sum of the elements of the diagonal is zero, then trace(A) = 0;

A tilde as element means that these elements may have any value.

Exponentials of Nilpotent Matrices

A matrix that becomes zero after being squared n times is called nilpotent of order n.

Nilpotent 2x2 Matrices

The square of the matrix A, with the element a in the upper right corner is the zero matrix.

Because of the nilpotent property of the matrix A the exponetial serie of e^A becomes zero after the first two elements. Hence the exponential of the matrix A is the summe of the first two elements of the series, the identitiy matrix and the element A:

e^A = I + A;

The square of the matrix B, with the element b in the lower left corner is the zero matrix.

Because of the nilpotent property of the matrix B the exponetial serie of e^B becomes zero after the first two elements. Hence the exponential of the matrix B is the summe of the first two elements of the series, the identitiy matrix and the element B:

e^B = I + B;

Product of the two nilpotent matrices A and B:

Product of the two nilpotent matrices B and A:

AB ≠ BA;

Product of the exponential matrix functions of A and B:

Special Case I

The matrix A is nilpotent. Aⁿ = 0 for n>1. The exponential series of e^A is zero after the second element. Hence e^A = I + A.

Special Case II

The matrix B is nilpotent. Bⁿ = 0 for n>1. The exponential series of e^B is zero after the second element. Hence e^B = I + B.

Hyperbolic Functions - Series Expansion

The taylor series of real valued functions e^a, cosh(a), sinh(a), ah₃(a), bh₃(a), ch₃(a) are generalised to matrix valued functions. The matrix A is a square matrix. The taylor series of the other functions are subseries of the taylor series of the exponential function.

From the taylor series definition of the functions we get the following relations between the defined functions:

e^A = cosh(A) + sinh(A) = ah₃(A) + bh₃(A) + ch₃(A);

Trigonometric Functions - Series Expansion

e^iA = cos(A) + isin(A) = ae₃(A) + be₃(A) + ce₃(A)

Cyclic Group C³

Basic Matrix Representation of the Cyclic Group C³

For the matrix representation of the cyclic group c³ by 3x3 matrices we start with two sets of exchange matrices where each set contains three exchange matrices.
Summing up the three matrices of a single set will result in a matrix that has a one at each possition.

First Set of Matrices:

Determinant

Multiplication Table

		I	A	B
I		I	A	B
A		A	B	I
B		B	I	A

Algebraic Relations

A² = B;

B² = A;

From the products we can identifiy the inverse elements of A and B.

AA^-1 = AB = I;   BB^-1 = BA = I

A^-1 = B;   B^-1 = A;

A³ = AAA = BA = I;

B³ = BBB = AB = I;

A⁴ = AAAA = BB = A;

B⁴ = BBBB = AA = B;

Commutator:

[A, B] = AB - BA = I - I = 0;

Anticommutator

{A, B} = AB + BA = I + I = 2I;

Roots of I

The matrices I, A, B are communative.

Renaming of the Matrices I, A, B

u₀ = I

u₁ = A

u₂ = B

The matrices u₀, u₁, u₂ are communative.

u₀² = u₀ = I = I²

u₁² = u₂

u₂² = u₁

u₁³ = I

u₂³ = I

Building the weighted summe of these matrices:

Performing the exponatiation of the weighted summe of the matrices:

e^{au₀ + bu₁ + cu₂} = e^au₀e^bu₁e^cu₂

Roots of I Definition

The matrices with the property Uⁿ = I are called roots ouf I, even if U ≠ I.

The identity matrix I = Iⁿ is mapped to itself for any integer value of n. For nxn matrices there are other matrirces with Uⁿ = I and U ≠ I.

Hyperbolic Functions of 3x3 Matrices

Definition of a 3x3 matrix M as roots of I.

From the taylor series of the exponential function the hyperbolic functions can be defined as subseries of the exponential function. Giving the following basic relation.

e^M = e^{u₀a + u₁b + u₂c} = ah₃(M) + bh₃(M) + ch₃(M)

Further exploration of the taylor series of the three hypebolic functions reveals further relations.

From the sum of these series we get the following basic relations.

Exponentiation

Using the substitution r = e^a;

Perform the following renaming gives the basic relations:

Power Series and Unit Roots

e^u₀a = ah₃(u₀a) + bh₃(u₀a) + ch₃(u₀a);

e^u₁b = ah₃(u₁b) + bh₃(u₁b) + ch₃(u₁b);

e^u₂c = ah₃(u₂c) + bh₃(u₂c) + ch₃(u₂c);

The unit roots u₀, u₁, u₂ have the properties:

(u0)n	=	u0;
(u1)3n	=	u0;
(u2)3n	=	u0;
(u1)2+3n	=	u2;
(u2)2+3n	=	u1;

Applying these properties to the power series will result in further relations.

This will give the following matrix representation:

Simple Trigonometric Function Matrices

From the power series above we can extract two matrices that contain simple trigonometric functions.

With e^a = r we have:

Compound Trigonometric Functions

The product of the two matrices with simple trigonometric fucntions will result in a matrix that contains the compound trigonometric functions:

Excecuting the product will give the defining relations:

Comparing the elements of the matrices will give the following relations between the simple trigonometric functions and the compound trigonometric functions:

Second Set of Matrices

Basic Matrices:

Second Set of Matrices:

Determinant

Multiplication Table

		I	A	B		C	D	E
I		I	A	B		C	D	E
A		A	B	I		E	C	D
B		B	I	A		D	E	C
C		C	D	E		I	A	B
D		D	E	C		B	I	A
E		E	C	D		A	B	I

Algebraic Relations

From the basic matrices from the second set the basic matrices of the first set can be constructed.

This gives the basic algebraic relations between the matrices:

I² = AB = BA = C² = D² = E² = I;

DE = A; ED = B;

A^-1 = B; B^-1 = A;

DE^-1 = A^-1 = B = ED;

ED^-1 = B^-1 = A = DE;

Commutator:

Anticommutator:

Negative Elements in the Basic Matrices

Basic Matrices:

A³ = AAA = BA = -I

B³ = BBB = -AB = I

Second Version

Basic Matrices:

A³ = AAA = -BA = -I

B³ = BBB = -AB = -I

Third Version

Basic Matrices:

A³ = AAA = -BA = I

B³ = BBB = AB = -I

4x4 Matrix Representation

Combination of a euclidien and a hyperbolic rotation. This is not a 4-dimensional rotation but a rotation in a euclidien plane and a rotation in a hyperbolic plane. These are two 2-dimensional rotations of a different kind.

Trace of a Matrix

The trace of a nxn matrix is defined as the sum of the diagonal elements.

Trace of the Product of Two Matrices

For the trace of the product of two matrices we have the following relation:

because

for a nxn matrix.

Trace of a Commutator

The trace of a commutator is zero.

Similar Matrices

Similar matrices have the same trace.

for an invertiable matrix B because:

Next Page:

Abstract Algebra