Wacker Art Linear Algebra Wappen der Familie Wacker
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Prolog

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Matrices

A matrix A is a rectangular array of aij elements. The index values i and j define the position of the element in the matrix. The i value defines the row value and the j value defines the coulumn value. A matrix can have m rows and n columns:

By default the elements aij of a matrix are real numbers. If the elements are from a different kind, this will be stated, or it becomes clear from the context in which the matrix is used. Sometimes the following shortform for a matrix is used:

A = aij

The element aij with variable indices ij is used as Symbol for all elements of a matrix.

Matrix Examples:

Special Matrices

Column Vector

Matrices with only one column are called column vectors.

Row Vector

Matrices with only one row are called row vectors.

Vectors and Matrices

A general vector can be written in the form a = ai. The information if it is a column vector or a row vector is only neccessary if the vector elements are interpreted as a matrix.

Quadratic Matrices

If the number of rows and the number of columns of a matirix are the same, the matrix is called a quadratic matrix:

Unit Matrix

A unit matrix In is a quadratic matrix where all the diagonal elements are one and all the other elements are zero.

Kronecker Delta

The Kronecker Delta is a function that depends on its two indices as variables.

The elements of a Unit Matrix can be defined with the Kronecker Delta. The Kronecker Delta defines an matrix element value depending on the element position by the following rules:

Kronecker Delta Function


Where i and j define the position of the matrix element.

In the case of a quadratic matrix the result of the Kronecker Delta function will be the unit matrix.

Single Entry Matrix

A single entry matrix eij is a matrix where a single element eij is one and all the other matrix elements are zero.

Row-Addition Transformation

Lij(a) = I + aeij (i≠j)

Row-Switching Transformation

Tij = I + eij + eji - eii - ejj

Row-Multiplying Transformation

Di(a) = I + (a-1)eii (a≠0)

Generalising Vectors and Matrices to Tensors

The rank value defines the number of indices an element a can have. An element without index is a scalar. The general name for an element with multiple indices is tensor.

Rank Element Name Tensor
0 a scalar tensor rank 0
1 ai vector tensor rank 1
2 aij matrix tensor rank 2
3 aijk tensor rank 3
... aijk... tensor rank ...

Basic Matrix Operations

Matrix Additon and Subtraction

When two matrices have the same number of lines and columns it is possible to perform the additon and the subtraction of two matrices. In this case the sum of two matrices is build by adding the matrix elements and the difference is build by subtracting the matrix elements.

A+B = aij + bij;    A-B = aij - bij;

Example:

Multiplication with a Scalar

The multiplication of a Matrix A with a scalar λ is the same as multipling each element aij of the matrix with a scalar λ.

λA = λaij

Example:

The Transposed of a Matrix

The transposed of a matrix is defined by echanging the columns and the rows of a given matrix.

AT = (aij)T = aji

Examples:

The transposed of a matrix is a matrix where the elements of the original matrix are mirrored at the diagonal:

Applying the operation of a transposition twice on a matrix, will result in the original matrix.

ATT = (aij)TT = (aji)T = aij = A

Matrix Product

The matrix product is not commutative. If A and B are matrices, then the product ABBA. The matrix product is only defined if the number of columns of the first matrix A is the same as the number of rows of the second matrix B. The product of the matrixes A and B defines a new matrix AB=C. The elements cij of the matrix C are defined by the following formular:

Matrix Product


Special Matrix Products:

Matrix product of the the row vector A with the column vector B where both vectors A and B have two elements.

The Product of a row vector A with a column vector B will be the Matrix C which has only one element called c11.

AB=C=(c11)=(a1b1+a2b2);    c11=a1b1+a2b2;

The result is the scalar product of the two vectors.

The product of a column vector with a row vector BA=D.

The result of the matrix product BA is a 2×2 matrix.

Redefining A and B to row and column vectors with 3 elements:

AB=C=(c11)=(a1b1+a2b2 + a3b3);    c11=a1b1+a2b2+ a3b3;

The result of the matrix product BA is a 3×3 matrix.

Examples - Matrix Products:

Product of the matrices A and B.

Permutation

In general a permutation is a rearrangement of elements of an ordered list like the elements of a vector. Permutations can be expressed as matrix operations on a vector. If an arrangement contains n elements n! different permutations are possible.

Permutaions as Matrix Operations

A permutation is a matrix operation that leads two the exchange of the elements of a vector, when the matrix is multiplied with the vector.

The unit matrix is a permutation matrix, that leaves the order of the vector elements unchanged when applied to the vector.

Exchange Matrix

The exchange Matrix is the next example of a permutation matrix.

The exchange matrix or counterbrace matrix Jn, is a matrix where the counterdiagonal elements are all one. All other elements are zero.

The product of the exchage matrix with a column vector will result in a reordering of the column elements. The product of a row vector with a exchange matrix will result in a reordering of the row elements.

Square of the Exchange Matrix

J2 = I

The square of the exchange matrix is the unit matrix. Applying the exchange matrix twice will result in the original matrix.

Transpositions

Transpositions are permutations where only two rows of a matrix are exchanged. Every permutation can be expressed as a product of transposions.

Transpositions with a 2*2 matrix:

With two elements 2! = 2 permutations are possible.

Transpositions with a 3*3 matrix

With three elements 3! = 6 permutations are possible.

Matrixes as Complex Units

Geometric Numbers

Geometric numbers comprise the real numbers and two new elements a,b with special properties. The new geometric numbers a,b also called g-numbers have the following properties:

(i)  a2=0=b2 The new g-numbers are called nilpotent or null vectors.

(ii) 2ab =ab+ba=1

The g-Numbers Expressed as Matrices

The canonic elements a and b expressed as matrices:

Squaring of the canonic matrices will result in the zero matrix:


a2 = 0;

b2 = 0;

Building the product a and b and the product b and a.

ab

ba

ab + ba = 1

aba=(ab)a=(1-ba)a=a-ba2=a

(ab)2=(1-ba)ab=ab+ba2b=ab

(ba)2=(1-ab)ba=ba+ab2a=ba

Levi-Civita-Symbol

The Levi-Civita-Symbol also known as the total antisymetric tensor or ε-tensor is a function of the values of its indices.

A specific Levi-Civita-Symbol for a specific permutation of its indices has to be defined for a specific value n. And has then n indices, where each index values can lay in a range between 1 and n.

The exchange of two index elements is called a transposition. The exchange of two index elements will result in a change of the sign of the epsilon tensor.

The general form of the εij...k tensor is defined as follows, it also holds if two indices are the same (none distinctive elements).

Levi-Civita-Symbol

A permutation is even if a even number of index transpositions has to be executed to get the permutation of 1,2...,n.

A permutation is odd   if a  odd number of index transpositions has to be executed to get the permutation of 1,2...,n.

If t is the number of transpositions to get a a specific permutation of distinctive index elements, then: εij...k = (-1)t

Example:

In 2-Dimensions the possible values can be presented in matrix form:

Levi-Civita-Symbol

In 3-Dimensions the result can be thought of as staging of three matrices.

Levi-Civita-Symbol

For further relations between the Levi-Civita-Symbol and geometric algebra visit my geometric algebra page.

Determinant

A determinant is a linear function that calculates a scalar value from the elements of a quadratic matrix.

Algorithems for calculating the value of a Determinant

The determinant of a one dimensional matrix contains only one element. The result of the determinant operation is the value of this element:

The determinant of a two dimensional matrix is calculated as follows:

The determinant of a three dimensional matrix can be developed after the first Line:

Also the determinant of a four dimensional matrix can be developed after the first Line:

In this manner a determinant of any dimension can be developed after the first line by buliding the determinants of the minor matrices.

Determinant Examples:

Minors

A submatrix Aij of a matrix A is the matrix that is received by deleting the ith row and the jth column of the matrix A.

Example:

The minor Mij = |Aij| is the determinant of the submatrix Aij.

Example:

Cofactors

The cofactor Cij is obtained by multiplying the minor Mij by (-1)i+j.

Cij=(-1)i+jMij


Examples:

The signum of a 4 x 4 matrix:


The Cofactor matrix of a 3 x 3 matrix:

The Adjugate Matrix or Classical Adjoint Matrix

The adjugate matrix adj(A) of a quadratic matrix A is the transposed of the cofactor matrix: adj(A) = CT = (-1)j+iMji

The Invers of a Matrix

The inverse of a quadratic matrix A is denoted as A-1. If a inverse matrix is defined, it shall have the following propertie:

AA-1 = 1

The product of a matrix A with its inverse matrix A-1 shall be the unit matrix 1.

When the adjungated matrix Aadj of a matrix A and the dererminant det(A) of a matrix A is given, the invers of a matrix A can be calculated by dividing each element of the adjungated matrix Aadj by the value of the determinant det(A):

Inverse Matrix


Not every matrix is inveritable. A matrix that has no inverse matrix is called singular or non regular. The above formular requires that det(A) ≠ 0.

Example:

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The next step is Geometric Algebra!

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18. Dezember 2019 Version 2.0
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