intro. to mathematical operation

7/31/2019 Intro. to Mathematical Operation

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Introduction to

Mathematical Modelling


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MATRIX OPERATION


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Multiplication of vectors

The matrix product of a row vector u with a column vectorw is defined in a same way as the vector dot product, that is:

If each vector has n element. Thus the result of multiplying a 1

X n vector times an n X 1 vector is a 1 X 1 array, that is a

scalar.

= 32


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Using MATLAB

>> a=[1 2 3]>> b=[4;5;6]

>> a*b

ans =

32

>> a=[1 2 3]

>> b=[4 5 6]

>> a*b

??? Error using ==> mtimes

Inner matrix dimensions must agree.


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Vector-matrix multiplication

Scalar result of each row-column multiplication forms an elementin the result, which is column vector :

For example :

The result of multiplying a 2x2 matrix times a

2x1 vector is a 2x1 array; that is a column vector.

==

=


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Using MATLAB

>> c=[2 7;6 -5]

>> d=[3;9]

>> e=c*d

e =

69

-27


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Matrix-Matrix Multiplication

In the product of two matrices AB, the number ofcolumnsin A must equal the number of rowsin B. Therow-column multiplications form column vectors, andthese column vectors form the matrix result. Theproduct AB has the same number of rowsas A and the

same number of columnsas B. For example,

6 2

10 3

4 7

9 8

5 12=

(6)(9) + ( 2)( 5) (6)(8) + ( 2)(12)

(10)(9) + (3)( 5) (10)(8) + (3)(12)

(4)(9) + (7)( 5) (4)(8) + (7)(12)

64 24

75 116

1 116

=


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Use the operator * to perform matrix multiplication inMATLAB. The following MATLAB session shows how to

perform the matrix multiplication shown in (2.44).

>>A = [6,-2;10,3;4,7];

>>B = [9,8;-5,12];

>>A*B

ans =

64 24

75 116

1 116


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Matrix multiplication does not have the commutativeproperty; that is, in general, AB BA.A simple

example will demonstrate this fact:

AB =6 2

10 3

9 8

12 14=

78 20

54 122

BA =9 8

12 14

6 2

10 3=

134 6

68 65

whereas

Reversing the order of matrix multiplication is acommon and easily made mistake.


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Matrix Division Matrix division uses both the right and left division operators, /

and \ We can add, subtract, multiply and exponentiate matrices, but we

cannot divide two matrices.

We might define matrix division as a process to do the same.

xy= z

Simple algebra tells us that dividing both sides by y yields a truestatement for the same variables:

x= z/ y ; Which is the same as :x=z*1/y

Or, xequals z times the inverse ofy. So matrix division can bedefined as multiplying both sides of an equation by the inverseof a matrix. For two by two matrices this should be as follows:

AB= C becomes B= (A -1)C


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Matrix notation enables us to represent multiple equationsas a single matrix equation. For example, consider the

following set:2x1+ 9x2= 5

3x1- 4x2= 7

This set can be expressed in vector-matrix form as

which can be represented in the following compact form

Ax = b

2 93 -4

x1x2

=57


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For the equation set Ax = b,

if |A| = 0, then there is no unique solution.

Depending on the values in the vector b, theremay be no solution at all, or an infinite number ofsolutions.

We can compute |A| by command det(A)= -35or by calculateEg. if A= = (2)(-4) - (3)(9) = -352 9

3 -4


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MATLAB provides the left-divisionmethod for solving the

equation set Ax = b. The left-division method is based onGauss elimination.

To use the left-division method to solve for x, type x = A\b.

For example,6 -10 X1 = 2

3 -4 X2 5

>> A = [6, -10; 3, -4]; b = [2; 5];

>> x = A\b

x =

7 4

This method also works in some cases where the number ofunknowns does not equal the number of equations.

Left Division Method


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If the number of equations equals the number ofunknowns and if |A| 0, then the equation set

has a solution and it is unique.

If |A| = 0 or if the number of equations does notequal the number of unknowns, then you mustuse the methods presented in Section 6.4.

6-11


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Matrix Inverse

The MATLAB command inv(A) computes the inverse of

the matrix A. The following MATLAB session solves thefollowing equations using MATLAB.

2x+ 9y= 53x- 4y= 7

>>A = [2,9;3,-4];

>>b = [5;7]

>>x = inv(A)*b

x =

2.37140.0286

If you attempt to solve a singular problem using the inv

command, MATLAB displays an error message.


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Underdetermined Systems

An underdetermined systemdoes not containenough information to solve for all of the unknownvariables, usually because it has fewer equationsthan unknowns.

Thus an infinite number of solutions can exist,with one or more of the unknowns dependent onthe remaining unknowns.

For such systems the matrix inverse method andCramers method will not work.


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A simple example of an underdetermined systems isthe equation

x+ 3y= 6

All we can do is solve for one of the unknowns interms of the other; for example, x= 6 - 3y. An infinitenumber of solutions satisfy this equation.

When there are more equations than unknowns, theleft-division method will give a solution with some ofthe unknowns set equal to zero. For example,>>A = [1, 3]; b = 6;

>>solution = A\b

solution =

0

2

which corresponds to x= 0 and y= 2.


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Special Product

Syntax

C = cross(A,B)

C = cross(A,B,dim)Description

C = cross(A,B) returns the cross product of the vectors A and B.That is, C = A x B. A and B must be 3-element vectors. If A andB are multidimensional arrays, cross returns the cross product of

A and B along the first dimension of length 3. C =cross(A,B,dim) where A and B are multidimensional arrays,returns the cross product of A and B in dimension dim . A and Bmust have the same size, and both size(A,dim) and size(B,dim)must be 3.

Examples

The cross and dot products of two vectors are calculated as shown:

a = [1 2 3];

b = [4 5 6];

c = cross(a,b)

c = -3 6 -3


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Special ProductSyntax

C = dot(A,B)

C = dot(A,B,dim)

DescriptionC = dot(A,B) returns the scalar product of the vectors A and B. A

and B must be vectors of the same length. When A and B areboth column vectors, dot(A,B) is the same as A'*B.

For multidimensional arrays A and B, dot returns the scalar product

along the first non-singleton dimension of A and B. A and Bmust have the same size.

C = dot(A,B,dim) returns the scalar product of A and B in thedimension dim.

Examples

The dot product of two vectors is calculated as shown:a = [1 2 3];

b = [4 5 6];

c = dot(a,b)

c = 32


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Summary Vectors/Matrix

Matlab also gives you the following operation such as transpose(`),

Addition (+) and multiplication (*), multiplication (*) by

transpose (`), dot product, u.v and cross product, uxv.


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Operation Matlab command Mathemathical

tools/comment

Define two vectors

u and v

>> u=[1 2 3]

u=

1 2 3>>v=[2 4 6]

v=

2 4 6

Define a pair of vector

u=[1 2 3]

&v=[2 4 6]

Transpose (`)

Addition (+)

Subtraction (-)

>> u`

Ans =

12

3

>> v`

Ans=

2

4

6>> u+v

Ans=

3 6 9

>> u-v

Ans=

-1 -2 -3

Interchange the

elements between rows

and columnsThe dimension of both

matrices must same for

addition and

substraction.

u v = [ 1 2 3 ] [ 2

4 6]

1x31x3


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Multiplication (*) >> u * v

??? Error using==>*

The dimension dont

agreeu v = [ 1 2 3 ] [ 2 4 6]

1x3 1x3

Multiplication (*)

by transpose (`)

>> u*v`

Ans =

28

Taking the

transpose works

uv` = [ 1 2 3 ] 2

4

6

1x3 3x1

dot product, u.v >> dot(u,v)

Ans=

28

The dot product is the

same thing

cross product, uxv >> cross(u,v)

Ans=

0 0 0

The cross product

works only for 3-D

vectors

intro. to mathematical operation

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