انت هنا الان : شبكة جامعة بابل > موقع الكلية > نظام التعليم الالكتروني > مشاهدة المحاضرة

Systems of Linear Algebraic Equations

الكلية كلية العلوم للبنات     القسم قسم فيزياء الليزر     المرحلة 3
أستاذ المادة نزار سالم شنان الزبيدي       4/3/2011 6:16:10 AM

Chapter 1

 

 

Systems of Linear Algebraic Equations

 

 

1.1 Introduction:

 

 

       The static mechanical spring-mass system illustrated in figure 1.1 consists of three masses m1 to m3, having weight W1 to W3, interconnected by five linear springs K1 to K5.

 

 

 

 

 

Figure 1.1 Static mechanical spring-mass systems.

 

 

When the supporting forces F₁ to F₃ are removed, the masses move downward and reach a new static equilibrium configuration, denoted by x₁, x₂ and x₃  where x₁, x₂ and x₃ measured from original location of the corresponding masses.

 

Free-body diagrams of the three masses are presented at the bottom of figure 1.1 performing a static force balance on the three masses yields the following system of three linear algebraic equation:

 

 

(K₁ +K₂+K₃)x₁ - K₂x₂ - K₃ x3 = W₁                               (1.1a)

 

- K₂x1 + (K₂+K₄)x₂ – K₄x₃ = W₂                                (1.1b)

 

- K₃x₁ – K₄x₂ + (K₃+K₄+K₅)x₃ = W₃                          (1.1c)

 

 

The static mechanical spring-mass system illustrated in figure 1.1 is used as the example problem in this chapter to illustrated methods for solving systems of linear algebraic equations.

 

 

Let

 

K₁ = 40 N/cm

 

K₂ = K₃ = K₄ = 20 N/cm

 

K₅ = 90 N/cm

 

W₁ = W₂ = W₃ = 20 N

 

 

For these values, equation (1.1) becomes:

 

 

80x₁ – 20x₂ – 20x₃ = 20                                           (1.2a)

 

- 20x₁ + 40x₂ – 20x₃ = 20                                         (1.2b)

 

- 20x₁ - 20x₂ + 130x₃ = 20                                        (1.2c)

 

 

The solution to equation (1.2) is

 

x₁= 0.6cm,                 x₂=1.0cm,                        x₃=0.4cm

 

 

 

The equations may be linear or nonlinear chapter1 is devoted to the solution of systems of linear algebraic equations of the following form:

 

 

a₁₁x₁  + a₁₂x₂ + a₁₃x₃ + a_1nx_n = b₁                                    (1.3a)

 

a₂₁x₁  + a₂₂x₂ + a₂₃x₃ + a_2nx_n = b₂                                (1.3b)

 

………………………………….

 

a_n1x₁  + a_n2x₂ + a_n3x₃ + a_nnx_n = b_n                                 (1.3n)

 

 

The number of equations can ranged from two to hundreds, thousands and even millions.

 

We can call the equations ranged like equation (1.3) a MATRIX.

 

1.2 Properties of Matrices and Determinants.

 

 

 

1.2.1 Matrix Definitions.

 

 

      A matrix is a rectangular array of elements (either number or symbols) which are arranged in orderly rows and columns. Each element of the matrix is distinct and separate. The location of an element in the matrix is important. Elements of matrix are generally identified by a double subscripted lowercase letter, for example, a_ij where the first subscript i identifies the row of matrix and second subscript j identifies the column of the matrix.

 

 

 

 

 

 

Comparing tow equation (1,3) and (1.5) shows that the coefficients of a system of linear algebraic equations from the elements of an nxn matrix.

 

 

Vectors are a special type of matrix which has only one column or one row.

 

A column vector is an nx1 matrix. Thus,

 

 A row vector is a 1xm matrix, for example

 

 

Y = [y_j] = [y₁  y₂   ………..y_m]      (j=1, 2,…………….m) ____(1.6b)

 

 

 

 

 

 

 

 

 

1.2.2 Matrix Algebra

 

 

       Matrix algebra consists of matrix addition, matrix subtraction and matrix multiplication. Matrix division is not defined. An analogous operation is accomplished using the matrix inverse.

 

 

Let A and B be two matrices of equal size. Thus

 

 

A+B = [a_ij] + [b_ij] = [a_ij +b_ij] = [C_ij] = C             ____________(1.7a)      

 

A - B = [a_ij] - [b_ij] = [a_ij - b_ij] = [C_ij] = C            ____________(1.7b)

 

 

Unequal size matrices cannot be added or subtracted.

 

 

A + (B+C) = (A+B) + C                                    _____________(1.8)

 

A + B = B + A                                                  ____________(1.9)

 

 

Example 1:

 

add the two 3x3 matrices A and B to obtain the 3x3 matrix C, where

 

 

 

 

 

Solution:

 

 

From Eq. (1.7)

 

c_ij = a_ij + b_ij

 

 

 

c₁₁ = a₁₁ + b₁₁ = 1 + 3 = 4

 

c₁₂ = a₁₂ + b₁₂ = 2+2 = 4, etc

 

 

 

 

 

 

 

 

 

Matrix multiplication consists of row-element to column-element multiplication and summation of the resulting products.

 

If the size of matrix A is n x m  and the size of matrix B is m x r then

 

 

  __(1.10)

 

 

Multiplication of the matrix A by the scalar ? consists of multiplying each element of A by ?. Thus

 

 

?A = ?[a_ij] = [?a_ij] = [b_ij] = B                           ______________(1.11)

 

 

Example 2

 

      Multiply the 3x3 matrix A and the 3x2 matrix B to obtain the 3x2 matrix C, where

 

 

 

 

 

Solution:

 

c₁₁ =  a₁₁b₁₁+a₁₂b₂₁+a₁₃b₃₁ = (1)(2)+(2)(1)+(3)(2) = 10

 

c₁₂ =  a₁₁b₁₂+a₁₂b₂₂+a₁₃b₃₂ = (1)(1)+(2)(2)+(3)(1) = 8

 

…………………………………………………………..

 

………………………………………………………….

 

………………………………………………………….

 

c₃₂ =  a₃₁b₁₂+a₃₂b₂₂+a₃₃b₃₂ = (1)(1)+(4)(2)+(3)(1) = 12

 

 

Thus

 

 

 

 

 

 

 

 

 

 

 

 

Transpose of matrix (A^T ) is obtained by inter changing the rows and the columns of A.

 

 

 If       then  

 

 

Example 3: transpose matrix A

 

 

 

 

 

Solution

 

 

 

Example 4: transpose matrix B  where    

 

Solution:   

 

 

 

Property:   (AB)^T = A^T B^T

 

 

 

The unit matrix is  

 

Where  I = I^T

 

 

Note: if we consider the two square matrices A and B and AB=I, then B is the inverse of A which is denoted as A^-1. Matrix commute on multiplication. Thus.

 

A A^-1 =A^-1 A = I