CSE123_Lecture4.pdf

CSE123

Lecture 4

Arrays,Matrices and Array

Operations

Definitions

• Scalars: Variables that represent single numbers. Note that complex numbers are also scalars, even though they have two components.

• Arrays: Variables that represent more than one number. Each number is called an element of the array. Array operations allow operating on multiple numbers at once.

• Row and Column Arrays (Vector): A row of numbers (called a row vector) or a column of numbers(called a column vector).

• Two-Dimensional Arrays (Matrix): A two-dimensional table of numbers, called a matrix.

Vector Creation by Explicit List

• A vector in Matlab can be created by an explicit list, starting with a left

bracket, entering the values separated by spaces (or commas) and

closing the vector with a right bracket.

>>p = [3,7,9]

p =

3 7 9

You can create a column vector by using the transpose

notation (').

>>p = [3,7,9]'

p =

3

7

9


You can also create a column vector by separating the

elements by semicolons. For example,

>>g = [3;7;9]

g =

3

7

9


>>x=[0 .1*pi .2*pi .3*pi .4*pi .5*pi .6*pi .7*pi .8*pi .9*pi pi]

>>y=sin(x)

>>y =

Columns 1 through 7

0 0.3090 0.5878 0.8090 0.9511 1.0000 0.9511

Columns 8 through 11

0.8090 0.5878 0.3090 0.0000

Vector Addressing / indexation

• A vector element is addressed in Matlab with an integer index (also called a subscript) enclosed in parentheses.

>> x(3)

ans =

0.6283

>> y(5)

ans =

0.9511

Colon notation: Addresses a block of elements. The format is:

(start:increment:end) Note start, increment and end must be positive integer numbers.

If the increment is to be 1, a shortened form of the notation may be used:

(start:end)

>> x(1:5)

ans =

0 0.3142 0.6283 0.9425 1.2566

>> x(7:end)

ans =

1.8850 2.1991 2.5133 2.8274 3.1416

>> y(3:-1:1)

ans =

0.5878 0.3090

>> y([8 2 9 1])

ans =

0.8090 0.3090 0.5878 0

Vector Creation Alternatives

• Combining: A vector can also be defined using another vector that has already been defined.

>> B = [1.5, 3.1];

>> S = [3.0 B]

S =

3.0000 1.5000 3.1000

• Changing: Values can be changed by referencing a specific address

• Extending: Additional values can be added using a reference to a specific

address.

>> S(2) = -1.0;

>> S

S =

3.0000 -1.0000 3.1000

>> S(4) = 5.5;

>> S

S =

3.0000 -1.0000 3.1000 5.5000

>> S(7) = 8.5;

>> S

S =

3.0000 -1.0000 3.1000 5.5000 0 0 8.5000


• Colon notation:

(start:increment:end)

where start, increment, and end can now be floating point numbers.

x=(0:0.1:1)*pi

x =

Columns 1 through 7

0 0.3142 0.6283 0.9425 1.2566 1.5708 1.8850


2.1991 2.5133 2.8274 3.1416

• linspace: generates a vector of uniformly incremented values, but instead of

specifying the increment, the number of values desired is specified. The form:

linspace(start,end,number)

1

number

startendincrement

The increment is computed internally, having the value:


>> x=linspace(0,pi,11)

x =

Columns 1 through 7

0 0.3142 0.6283 0.9425 1.2566 1.5708 1.8850


2.1991 2.5133 2.8274 3.1416

logspace(start_exponent,end_exponent,number)

To create a vector starting at 100 = 1,ending at 102 = 100 and having 11 values:

>> logspace(0,2,11)

ans =

Columns 1 through 7

1.0000 1.5849 2.5119 3.9811 6.3096 10.0000 15.8489


25.1189 39.8107 63.0957 100.0000

Length, and Absolute Value of a Vector

Keep in mind the precise meaning of these terms when

using MATLAB.

The length command gives the number of elements in

the vector.

The absolute value of a vector x is a vector whose

elements are the absolute values of the elements of x.

2-9

For example, if x = [2,-4,5],

its length is 3; (computed from length(x))

its absolute value is [2,4,5] (computed

from abs(x)).

2-10

Matrix arrays in Matlab

Vector •Use square brackets.

•Separate elements on the same row with spaces or

commas.

&

Matrix

•Use semi-colon to go to the next row.

A= [ 1 2 3 ];

C= [ 5 ; 6 ; 7 ];

321A

43

21B

7

6

5

C

B= [ 1 2 ; 3 4 ];

A= [ 1, 2, 3 ];

Matrix Arrays

A matrix array is 2D, having both multiple rows and multiple columns.

Creation of 2D arrays follows that of row and column vectors:

– Begin with [ end with ]

– Spaces or commas are used to separate elements in a row.

– A semicolon or Enter is used to separate rows.

>> h = [1 2 3

4 5 6

7 8 9]

h =

1 2 3

4 5 6

7 8 9

>> k = [1 2;3 4 5]

??? Number of elements in each

row must be the same.

>>f = [1 2 3; 4 5 6]

f =

1 2 3

4 5 6

>> g = f’

g =

1 4

2 5

3 6

Manipulations and Combinations:

>> A=10*ones(2,2) A = 10 10

10 10

Special matrix creation

>> B=10*rand(2,2)

Matrix full of 10:

Matrix of random

numbers between

0 and 10

B = 4.5647 8.2141

0.1850 4.4470

>> C= -rand(2,2)

Matrix of random

numbers between

-1 and 0

C = -0.4103 -0.0579

-0.8936 -0.3529

>> C=2*rand(2,2) –ones(2,2)

>> C=2*rand(2,2) -1

Matrix of random

numbers between

-1 and 1

C = -0.6475 0.8709

-0.1886 0.8338

Concatenation: Combine two (or more) matrices into one


Notation:

C=[ A, B ]

Square

brackets

>> A=ones(2,2);

>> B=zeros(2,2);

>>C=[A , B]

>>D=[A ; B]

D = 1 1

1 1

0 0

0 0

C = 1 1 0 0

1 1 0 0

Obtain a single value from a matrix:

Ex:

want to know a21

Matrix indexation

421

123

321

A

Notation:

A(2,1)

Row index Column index

>> A=[1 2 3; 3 2 1; 1 2 4];

>> A(2,1)

ans =

3

>> A(3,2)

ans =

2

Obtain more than one value from a matrix:

Ex: X=1:10

Matrix indexation

>> A=[1 2 3; 3 2 1; 1 2 4];

>> B=A(1:3,2:3)

B = 2 3

2 1

2 4

>> C=A(2,:)

C = 3 2 1

Colon

Colon defines a

“range”: 1 to 10

Notation:

A(1:3,2:3)

Row 1 to 3

Column 2 to 3

Colon can also be used as a “wildcard”

Row 2, ALL

columns

421

123

321

A

Matrix size

Command Description

s = size(A) For an m x n matrix A, returns the two-element row

vector s = [m, n] containing the number of rows and

columns in the matrix.

[r,c] = size(A) [r,c] = size(A) Returns two scalars r and c

containing the number of rows and columns in A,

respectively.

r = size(A,1) Returns the number of rows in A in the variable r.

c = size(A,2) Returns the number of columns in A in the variable c.

Matrix size

>> A = [1 2 3; 4 5 6]

A =

1 2 3

4 5 6

>> s = size(A)

s =

2 3

>> [r,c] = size(A)

r =

2

c =

3

>> whos

Name Size Bytes Class

A 2x3 48 double array

ans 1x1 8 double array

c 1x1 8 double array

r 1x1 8 double array

s 1x2 16 double array

Additional Array Functions

max(A) Returns the algebraically largest element in A if

A is a vector.

Returns a row vector containing the largest elements in each column if A is a matrix.

If any of the elements are complex, max(A)

returns the elements that have the largest

magnitudes.

2-18

Additional Array Functions

[x,k] =

max(A)

min(A)

and

[x,k] =

min(A)

Similar to max(A) but stores the maximum values in the row vector x and their indices in the row vector k.

Like max but returns minimum values.

2-19

2-20

sort(A) Sorts each column of the

array A in ascending order

and returns an array the same size as A.

sum(A) Sums the elements in each

column of the array A and

returns a row vector

containing the sums.

Additional Array Functions (Table 2.1–1)

zeros(M,N) Matrix of zeros

ones(M,N) Matrix of ones

eye(M,N) Matrix of ones on the

diagonal

rand(M,N) Matrix of random

numbers between 0

and 1

>> A=zeros(2,3)

A = 0 0 0

0 0 0

>> B=ones(2,2)

B = 1 1

1 1

>> D=rand(3,2)

D = 0.9501 0.4860

0.2311 0.8913

0.6068 0.7621


>> C=eye(2,2)

C = 1 0

0 1

Operations on vectors and matrices in Matlab

Math Matlab

Addition/subtraction A+B

A-B

Multiplication/ division

(element by element)

A.*B

A./B

Multiplication

(Matrix Algebra)

A*B

Transpose: AT A’

Inverse: A-1 inv(A)

Determinant: |A| det(A)

“single quote”

Array Operations

Scalar-Array Mathematics

Addition, subtraction, multiplication, and division of an array by a scalar

simply apply the operation to all elements of the array.

>> f = [1 2 3; 4 5 6]

f =

1 2 3

4 5 6

>> g = 2*f -1

g =

1 3 5

7 9 11

Array Operations

Element-by-Element Array-Array Mathematics

When two arrays have the same dimensions, addition, subtraction, multiplication, and division apply on an element-by-element basis.

Operation Algebraic Form Matlab

Addition a + b a + b

Subtraction a − b a - b

Multiplication a x b a .* b

Division a / b a ./ b

Exponentiation ab a .^ b

333231

232221

131211

aaa

aaa

aaa

A

333231

232221

131211

bbb

bbb

bbb

B

333332323131

232322222121

131312121111

bababa

bababa

bababa

BA

MATRIX Addition (substraction)

M

N

M

N

M

N

Array Operations

987

654

321

A

BA

Examples: Addition & Subtraction

987

654

321

B

2 4 6

8 10 12

14 16 18

BA

0 0 0

0 0 0

0 0 0

Array Operations

Array Operations

Element-by-Element Array-Array Mathematics

>> A = [2 5 6];

>> B = [2 3 5];

>> C = A.*B

C =

4 15 30

>> D = A./B

D =

1.0000 1.6667 1.2000

>> E = A.^B

E =

4 125 7776

>> F = 3.0.^A

F =

9 243 729

333231

232221

131211

aaa

aaa

aaa

A

333231

232221

131211

bbb

bbb

bbb

B

333332323131

232322222121

131312121111

bababa

bababa

bababa

BA

MATRIX Multiplication (element by element)

M

N

BA

M

N

M

N

“dot”

“multiply”

NOTATION

Array Operations

987

654

321

A

BA

Examples: Multiplication & Division

(element by element)

987

654

321

B

1 4 9

16 25 36

49 64 81

B/A

1 1 1

1 1 1

1 1 1

Array Operations

The built-in MATLAB functions such as sqrt(x) and

exp(x) automatically operate on array arguments to

produce an array result the same size as the array argument x.

Thus these functions are said to be vectorized functions.

For example, in the following session the result y has

the same size as the argument x.

>>x = [4, 16, 25];

>>y = sqrt(x)

y =

2 4 5

2-33

However, when multiplying or dividing these functions, or

when raising them to a power, you must use element-by-

element operations if the arguments are arrays.

For example, to compute z = (ey sin x) cos2x, you must type

z = exp(y).*sin(x).*(cos(x)).^2.

You will get an error message if the size of x is not the same

as the size of y. The result z will have the same size as x

and y.

2-34 More? See pages 67-69.

We can raise a scalar to an array power. For example, if p = [2, 4, 5], then typing 3.^p produces the array

[32, 34, 35] = [9, 81, 243].

Remember that .^ is a single symbol. The dot in 3.^p

is not a decimal point associated with the number 3. The following operations, with the value of p given here, are

equivalent and give the correct answer:

3.^p

3.0.^p

3..^p

(3).^p

3.^[2,4,5]

2-38 More? See pages 70-72.

Array Operations

The matrix multiplication of m x n matrix A and nxp matrix B yields m x p matrix C, denoted by

C = AB

Element cij is the inner product of row i of A and column j of B

n

k

kjikij bac1

Note that AB ≠ BA

Matrix Multiplication

333231

232221

131211

aaa

aaa

aaa

A

333231

232221

131211

bbb

bbb

bbb

B

333323321331323322321231313321321131

332323221321322122221221312321221121

331323121311321322121211

bababababababababa

bababababababababa

babababababa

BA

Matrix Multiplication

Row 1

Colu

mn 1

311321121111 bababa

BA

“multiply”

NOTATION

M1

N1

M2

N2

M1

N2

N1=M2

Array Operations

213

123

321

A

BA

Example: Matrix Multiplication

16 9 11

12 11 13

12 10 14

213

123

321

B

1x1 + 2x3 +3x3 1x2 + 2x2 +3x1 1x3 + 2x1 +3x2

Array Operations

Solving systems of linear equations

Example: 3 equations and 3 unknown

1x + 6y + 7z =0

2x + 5y + 8z =1

3x + 4y + 5z =2

Can be easily solved by hand, but what can we do if it we have 10 or 100

equations?

Array Operations


First, write a matrix with all the (xyz)

coefficients

543

852

761

A

1x + 6y + 7z = 0

2x + 5y + 8z = 1

3x + 4y + 5z = 2

Write a matrix with all the constants

2

1

0

B

Finally, consider the matrix of unknowns

z

y

x

S

Array Operations


A x S = B

A x S = B A-1 x A-1 x

(A-1 x A) x S = A-1 x B

I x S = A-1 x B

S = A-1 x B

Array Operations


1x + 6y + 7z =0

2x + 5y + 8z =1

3x + 4y + 5z =2

The previous set of equations can be expressed in the following vector-matrix

form:

A x S = B

543

852

761

2

1

0

z

y

x

X

Array Operations

Matrix Determinant

Notation: Determinant of A = |A| or det(A)

•The determinant of a square matrix is a very useful value for finding if a system

of equations has a solution or not.

•If it is equal to zero, there is no solution.

det(M)= m11 m22 – m21 m12

2221

1211

mm

mmM

Formula for a 2x2 matrix:

IMPORTANT: the determinant of a matrix is a scalar

Array Operations

Matrix Inverse

Notation: inverse of A = A-1 or inv(A)

•The inverse of a matrix is really important concept, for matrix algebra

•Calculating a matrix inverse is very tedious for matrices bigger than 2x2.

We will do that numerically with Matlab.

2221

1211

mm

mmM M-1=

Formula for a 2x2 matrix:

1121

1222

)det(

1

mm

mm

M

IMPORTANT: the inverse of a matrix is a matrix

Array Operations

Property of identity matrix:

I x A = A

and A x I = A

Matrices properties

Property of inverse :

A x A-1 = I

and A-1 x A = I

Example:

100

010

001

2.04.08.0

2.06.02.0

6.02.04.0

x

102

121

211

Array Operations

x + 6y + 7z =0

2x + 5y + 8z =1

3x + 4y + 5z =2

543

852

761

A

2

1

0

B

z

y

x

S

In Matlab:

>> A=[ 1 6 7; 2 5 8; 3 4 5]

>> B=[0;1;2];

>> S=inv(A)*B

Verification:

>> det(A)

ans =

28

Solving systems of equations in Matlab

>>

S =

0.8571

-0.1429

0

x + 6y + 7z =0

2x + 5y + 8z =1

3x + 4y + 9z =2

943

852

761

A

2

1

0

B

z

y

x

S

In Matlab:

>> A=[ 1 6 7; 2 5 8; 3 4 5]

>> B=[0;1;2];

>> S=inv(A)*B

Verification:

>> det(A)

ans =

0

NO Solution!!!!!

Solving systems of equations in Matlab

Warning: Matrix is singular to working precision.

>>

S =

NaN

NaN

NaN

Applications in mechanical engineering

F1

F2 5N

7N

x

y

60o

30o

20o

80o

Find the value of the forces F1and F2

F1

F2 5N

7N

x

y

60o

30o

20o

80o

Projections on the X axis F1 cos(60) + F2 cos(80) – 7 cos(20) – 5 cos(30) = 0


F1

F2 5N

7N

x

y

60o

30o

20o

80o

Projections on the Y axis F1 sin(60) - F2 sin(80) + 7 sin(20) – 5 sin(30) = 0


F1 cos(60) + F2 cos(80) – 7 cos(20) – 5 cos(30) = 0

F1 sin(60) - F2 sin(80) + 7 sin(20) – 5 sin(30) = 0

F1 cos(60) + F2 cos(80) = 7 cos(20) + 5 cos(30)

F1 sin(60) - F2 sin(80) = - 7 sin(20) + 5 sin(30)

In Matlab:

>> CF=pi/180;

>> A=[cos(60*CF), cos(80*CF) ; sin(60*CF), –sin(80*CF)];

>> B=[7*cos(20*CF)+5*cos(30*CF) ; -7*sin(20*CF)+5*sin(30*CF) ]

>> F= inv(A)*B or (A\B)

F =

16.7406

14.6139

In Matlab, sin and cos use radians, not degree

Solution:

F1= 16.7406 N

F2= 14.6139 N


Polynomial Coefficients

The function poly(r)computes the coefficients of the

polynomial whose roots are specified by the vector r.

The result is a row vector that contains the polynomial’s

coefficients arranged in descending order of power.

For example,

>>c = poly([-2, -5])

c =

1 7 10

2-46

Polynomial Roots

The function roots(a)computes the roots of a polynomial

specified by the coefficient array a. The result is a

column vector that contains the polynomial’s roots.

For example,

>>r = roots([2, 14, 20])

r =

-2

-5

2-47

Polynomial Multiplication and Division

The function conv(a,b) computes the product of the two

polynomials described by the coefficient arrays a and b.

The two polynomials need not be the same degree. The

result is the coefficient array of the product polynomial.

The function [q,r] = deconv(num,den) computes the

result of dividing a numerator polynomial, whose coefficient array is num, by a denominator polynomial

represented by the coefficient array den. The quotient

polynomial is given by the coefficient array q, and the

remainder polynomial is given by the coefficient array r.

2-48

Polynomial Multiplication and Division: Examples

>>a = [9,-5,3,7];

>>b = [6,-1,2];

>>product = conv(a,b)

product =

54 -39 41 29 -1 14

>>[quotient, remainder] = deconv(a,b)

quotient =

1.5 -0.5833

remainder =

0 0 -0.5833 8.1667

2-49

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CSE123_Lecture4.pdf