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Prolog

Space, the final frontier.

The Exponential of a Matrix

Exponential Series

The matrix exponential eA means that the matrix is in the exponent. The matrix exponential is defined by a inifinit series 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 eA is a nxn matrix.

If A is a matrix then eA is a matrix.
If a is a real number then ea is a real number.

Commuative Matrices

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

eA+B = eAeB

Invers of a Exponential Matrix

e-AeA = eAe-A = eA-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:

PeAP-1 = ePAP-1

Trace of the Exponential Determinant

For quadratic matrices A we have:

det(eA) = etrace(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 eA 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:

eA = 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 eB 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:

eB = 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 eA is zero after the second element. Hence eA = I + A.

Special Case II

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

Hyperbolic Functions - Series Expansion

The taylor series of real valued functions ea, 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:

eA = cosh(A) + sinh(A) = ah3(A) + bh3(A) + ch3(A);

Trigonometric Functions - Series Expansion

eiA = cos(A) + isin(A) = ae3(A) + be3(A) + ce3(A)

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.

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:

eau0 + bu1 + cu2 = eau0ebu1ecu2

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.

eM = eu0a + 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 = ea;

Perform the following renaming gives the basic relations:

x3 + y3 + z3 - 3xyz = r3

Power Series

eu0a = ah3(u0a) + bh3(u0a) + ch3(u0a);

eu1b = ah3(u1b) + bh3(u1b) + ch3(u1b);

eu2c = ah3(u2c) + bh3(u2c) + ch3(u2c);

With the given relations we can construct the following power series:

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 ea = 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.

Passat
Bild: "Passat"

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31. Januar 2021 Version 2.0
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