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Space, the final frontier.
The matrix exponential e^{A} 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 e^{A} is a nxn matrix.
If A is a matrix then e^{A} is a matrix.
If a is a real number then e^{a} is a real number.
A and B are matrices with AB = BA (the matrices are commutative) then
e^{A+B} = e^{A}e^{B}
e^{-A}e^{A} = e^{A}e^{-A} = e^{A-A} = e^{0} = I;
If a matrix has only elements on the Diagonal we have:
If P is an invertibal nxn matrix and A is a nxn matrix we have:
Pe^{A}P^{-1} = e^{PAP-1}
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.
A matrix that becomes zero after being squared n times is called nilpotent of order n.
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:
The matrix A is nilpotent. A^{n} = 0 for n>1. The exponential series of e^{A} is zero after the second element. Hence e^{A} = I + A.
The matrix B is nilpotent. B^{n} = 0 for n>1. The exponential series of e^{B} is zero after the second element. Hence e^{B} = I + B.
The taylor series of real valued functions e^{a}, cosh(a), sinh(a), ah_{3}(a), bh_{3}(a), ch_{3}(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) = ah_{3}(A) + bh_{3}(A) + ch_{3}(A);
e^{iA} = cos(A) + isin(A) = ae_{3}(A) + be_{3}(A) + ce_{3}(A)
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. |
For the matrix representation of the cyclic group c^{3} 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.
I | A | B | ||
I | I | A | B | |
A | A | B | I | |
B | B | I | A |
A^{2} = B;
B^{2} = 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;
A^{3} = AAA = BA = I;
B^{3} = BBB = AB = I;
A^{4} = AAAA = BB = A;
B^{4} = BBBB = AA = B;
Commutator:
[A, B] = AB - BA = I - I = 0;
Anticommutator
{A, B} = AB + BA = I + I = 2I;
The matrices I, A, B are communative.
Renaming of the Matrices I, A, B
u_{0} = I
u_{1} = A
u_{2} = B
The matrices u_{0}, u_{1}, u_{2} are communative.
u_{0}^{2} = u_{0} = I = I^{2}
u_{1}^{2} = u_{2}
u_{2}^{2} = u_{1}
u_{1}^{3} = I
u_{2}^{3} = I
Building the weighted summe of these matrices:
Performing the exponatiation of the weighted summe of the matrices:
e^{au0 + bu1 + cu2} = e^{au0}e^{bu1}e^{cu2}
The matrices with the property U^{n} = I are called roots ouf I, even if U ≠ I.
The identity matrix I = I^{n} is mapped to itself for any integer value of n. For nxn matrices there are other matrirces with U^{n} = I and U ≠ I.
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} = ah_{3}(M) + bh_{3}(M) + ch_{3}(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.
Using the substitution r = e^{a};
Perform the following renaming gives the basic relations:
x^{3} + y^{3} + z^{3} - 3xyz = r^{3}
e^{u0a} = ah_{3}(u_{0}a) + bh_{3}(u_{0}a) + ch_{3}(u_{0}a);
e^{u1b} = ah_{3}(u_{1}b) + bh_{3}(u_{1}b) + ch_{3}(u_{1}b);
e^{u2c} = ah_{3}(u_{2}c) + bh_{3}(u_{2}c) + ch_{3}(u_{2}c);
With the given relations we can construct the following power series:
This will give the following matrix representation:
From the power series above we can extract two matrices that contain simple trigonometric functions.
With e^{a} = r we have:
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:
Basic Matrices:
Second Set of Matrices:
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 |
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:
I^{2} = AB = BA = C^{2} = D^{2} = E^{2} = 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:
Basic Matrices:
A^{3} = AAA = BA = -I
B^{3} = BBB = -AB = I
Basic Matrices:
A^{3} = AAA = -BA = -I
B^{3} = BBB = -AB = -I
Basic Matrices:
A^{3} = AAA = -BA = I
B^{3} = BBB = AB = -I
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.
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