computer programming eciv 2303 chapter 3 mathematical...

Computer Programming

ECIV 2303

Instructor: Dr. Talal Skaik

Islamic University of Gaza

Faculty of Engineering

Chapter 3

Mathematical Operations with Arrays

1Dr. Talal Skaik 2018

https://www.google.com/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&cad=rja&uact=8&ved=2ahUKEwj9ypP7hN7dAhWCNOwKHf_xB40QjRx6BAgBEAU&url=https://www.mathworks.com/help/matlab/examples/creating-the-matlab-logo.html&psig=AOvVaw2bzX-Ugh4SfSFdRmQg7yUg&ust=1538235616256324

3.1 ADDITION AND SUBTRACTION

In general, if A and B are two arrays (for example, 2 x 3 matrices),




When a scalar (number) is added to (or subtracted from) an array, the scalar is added to (or subtracted from) all the elements of the array.

3.2 ARRAY MULTIPLICATION


The operation A* B can be carried out only if the number of columns in matrix A is equal to

the number of rows in matrix B.

The result is a matrix that has the same number of rows as A and the same number of

columns as B.

For example, if A is a 4 x 3 matrix and B is a 3 x 2 matrix:

then the matrix that is obtained with the operation A*B has dimensions 4 x 2.



A*B



Numerical Example



Linear algebra rules of array multiplication provide a convenient way for writing a system of

linear equations. For example, the system of three equations with three unknowns

can be written in a matrix form as

and in matrix notation as

Identity Matrix


The identity matrix is a square matrix in which the diagonal elements are 1s and the rest of

the elements are 0s.

An identity matrix can be created in MATLAB with the eye command.

When the identity matrix multiplies another matrix (or vector), that matrix (or vector) is

unchanged.

Inverse of a Matrix


The matrix B is the inverse of the matrix A if, when the two matrices are multiplied, the

product is the identity matrix.

Both matrices must be square, and the multiplication order can be BA or AB.

BA = AB = I

The inverse of a matrix A is typically written as A-1 .

In MATLAB the inverse of a matrix can be obtained either by raising A to the power of -1,

A-1, or with the inv (A) function.

Inverse of a Matrix


Inverse of a Matrix


Not every matrix has an inverse. A matrix has an inverse only if it is square and its determinant is not equal to zero.

Determinants


A determinant is a function associated with square matrices.

The determinant is typically denoted by det(A) or |A|.

The determinant is calculated according to specific rules. For a second-order 2 x 2 matrix,

the rule is:

ARRAY DIVISION


MATLAB has two types of array division, right division and left division.

Left division, \ :Left division is used to solve the matrix equation AX = B.In this equation X and B are column vectors.

ARRAY DIVISION


Right division, / :The right division is used to solve the matrix equation XC = D.

In this equation X and D are row vectors.

Solving three linear equations (array division)


Use matrix operations to solve the following system of linear equations.

4x- 2y+ 6z = 8

2x+ 8y+ 2z = 4

6x+10y+3z = 0

SolutionThe above system of equations can be written in the matrix form AX = B or in the form

XC= D:

3.4 ELEMENT-BY-ELEMENT OPERATIONS


Element-by-element multiplication, division, or exponentiation of two vectors or matrices is

entered in MATLAB by typing a period in front of the arithmetic operator.

If two vectors a and b are a = [a1 a2 a3 a4] and b = [b1 b2 b3 b4], then element-by-element

multiplication, division, and exponentiation of the two vectors gives:



If two matrices A and B are

then element-by-element multiplication and division of the two matrices give:



Element-by-element exponentiation of matrix A gives:



Element-by-element calculations are very useful for calculating the value of a function at many values of its argument

3.5 USING ARRAYS IN MATLAB BUILT-IN MATH FUNCTIONS


if a vector with seven elements is substituted in the function cos (x), the result is a vector with seven elements in which each element is the cosine of the corresponding element in x.

3.5 USING ARRAYS IN MATLAB BUILT-IN MATH FUNCTIONS


An example in which the argument variable is a matrix is:

3.6 BUILT-IN FUNCTIONS FOR ANALYZING ARRAYS


Inverse of a 2x2 Matrix



Example:



Inverse of a 3x3 Matrix:

3. 7 GENERATION OF RANDOM NUMBERS


The rand command: The rand command generates uniformly distributed random numbers with values between 0 and 1.



Random numbers that are distributed in a range (a,b) can be obtained by multiplying rand

by (b- a) and adding the product to a:

(b- a)*rand +a

For example, a vector of 10 elements with random values between -5 and 10 can be created

by (a= -5, b = 10):



The randi command: generates uniformly distributed random integer.



The range of the random integers can be set to be between any two integers by typing [imin imax] instead of imax. For example, a 3 x 4 matrix with random integers between 50 and 90 is created by:



The randn command: generates normally distributed numbers with mean 0 and standard deviation of 1.

computer programming eciv 2303 chapter 3 mathematical...

Documents