Review of MATLAB

Review of MATLAB • The basics of MATLAB• Manipulating matrices• Built-in functions• Plotting• Functions in Matlab• Mathematical operations and system

of eqs.

NOTE: Covers Chapts. 1-6, but you should be pretty good if you mainly focus on the first 4 chapters

Slides most important: 1-48, 53-57, and 59

The Basic Data Type in MATLAB: Matrices

Group of numbers arranged into rows and columnsSingle Value (Scalar) Matrix with one row and one column

Vector (One dimensional matrix) One row or one column

Matrix (Two dimensional)

To create a row vector, enclose a list of values in brackets

You may use either a space or a comma as a “delimiter” in a row vector

Use a semicolon as a delimiter to create a new row

Shortcuts• While a complicated matrix might have to be

entered by hand, evenly spaced matrices can be entered much more readily. The command

b= 1:5

or the command

b = [1:5]

both return a row matrix

The default increment is 1, but if you want to use a different increment put it between the first and final values

To calculate spacing between elements use linspace

initial value in the array

final value in the array

number of elements in the array

“What’s the output”

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

This statement stores the matrix:

1. A=1 2 3

3 5 6

2. A= 1 2 3 3 5 6

3. A=

1 3

2 5

3 6

4. A= 1 2 3;3 5 6

What’s the output?

>>A = [0:0.1:0.5]


1. A= 0 0.1 0.2 0.3 0.4 0.5 2. A=

3. A= 4. A =

0

0 0.1 0.5 0.1 0.5


>>A = 5*[0:0.1:0.5]


1. A= 0 0.5 1.0 1.5 2.0 2.5 2. A=

3. A= 4. A =

0

0 0.1 0.5 0.1 0.5


>>A = linspace(0,10,3)


1. A= 0 5 10 2. A=

3. A= 4. A =

0 10 3

0 3 6 9 0 10 10

Manipulating Matrices

• Defining matrices• A matrix can be defined by typing in a list of numbers

enclosed in square brackets.• The numbers can be separated by spaces or

commas.• New rows are indicated with a semicolon.

A = [ 3.5 ];B = [1.5, 3.1]; or B =[1.5 3.1];C = [-1, 0, 0; 1, 1, 0; 0, 0, 2];


• Defining matrices• Define a matrix by using another matrix that has

already been defined.

• Reference an element in a matrix• Both row and column indices start from 1.

B = [1.5, 3.1]; S = [3.0, B]

S = [3.0, 1.5, 3.1]; T = [1 2 3; S]

S = 3.0 1.5 3.1 T = 1 2 3 3.0 1.5 3.1

S(2)

T(2, 2)

T(4)

The element value on row 2 and column 2 of matrix T

Count down column 1, then down column 2, and finally down column 2 to the correct element of matrix T.


• Change values in a matrix• S(2) = -1.0;

changes the 2nd value in the matrix S

from 1.5 to -1.0

• Extend a matrix by defining new elements.

S(4) = 5.5;Extend the matrix S to four elements instead of three.

S(8) = 9.5;Matrix S will have eight values, and the values of S(5), S(6), S(7) will be set to 0.

T(3, 3) = 10;T(4, 5) = 20;


• Using the colon operator• Define an evenly spaced matrix

H = 1:8 --- The default spacing is 1

time = 0.0:0.5:3.0 --- The middle value becomes the spacing.

• Extract data from matrices x = M( :, 1) --- extract column 1 from matrix M

y = M( :, 4) --- extract column 4 from matrix M

z = M(2, : ) --- extract row 2 from matrix M

a = M(2:3, : ) --- extract rows 2 and 3 from matrix M

b = M( :, 2:4) --- extract column 2 to column 4 from matrix M

c = M(2:3, 3:5) --- extract not whole rows or whole columns

from matrix M

• Converts a two dimensional matrix to a single column M( : )

M = 1 2 3 4 5 2 3 4 5 6 3 4 5 6 7

What’s does B store?

>>A = [5 6 ; 7 8]>>B = A(:,2)


1. B= 6 8

2. B= 7 8

3. B= 5 7 4. B= 7 8

NEXT TOPIC: MATLAB BUILT-IN FUNCTIONS

MATLAB uses function names consistent with most major programming languages

For example sqrt sin cos log

Function Input can be either scalars or matrices

Trigonometric Functions

sin(x) sinecos(x) cosinetan(x) tangentasin(x) inverse sinesinh(x) hyperbolic sineasinh(x) inverse hyperbolic sinesind(x) sine with degree inputasind(x) inverse sin with degree output

Data Analysis

max(x)min(x)mean(x)median(x)sum(x)prod(x)sort(x)

When x is a matrix, the max is found for each column

max value

element number where the max value occurs

Returns both the smallest value in a vector x and its location in vector x.

Vector of maximums

Vector of row numbers

Returns a row vector containing the minimum element from each column of a matrix x, and returns a row vector of the location of the minimum in each column of matrix x.

What does B store?

>>A = [1 2 3; 3 5 6;1 2 1]>>B = max(A)

c) 6 5 6

b) 3 6 2a) 6

d) 3 5 6

What does C store?

>>A = [1 2 3; 3 5 6;1 2 1]>>B = max(A)>>C=max(B)

c) 6 5 6

b) 3 6 2a) 6

d) 3 5 6

What does B and C store?

>>A = [1 2 3; 3 5 6;1 2 1]>>[B,C] = max(A)

c) B = 6 5 6 C = 1 1 1

b) B = 3 6 2 C = 3 3 2

a) B= 6 C = 4

d) B = 3 5 6 C = 2 2 2

What does D and E store?

>>A = [1 2 3; 3 5 6;1 2 1]>>[B,C] = max(A)>>[D,E] = max(B)

c) D = 6 E = 2

b) D = 6 E = 3

a) D= 6 E = 4

d) D = 3 5 6 E = 3

What is B?

>>A = [4 4 4 1 3 1 1]>>B = median(A)

c) B = 7

b) B = 3a) B = 4

d) B = 1

Will this piece of code work? If not, why?

>>A = [1 2 3; 3 5 6]>>B = [2 4 6; 6 10 18]>>C = B*A

This code will not work because ‘*’ is used for matrixmultiplication and in matrix multiplication you need the same number of columns in B as rows in A.

Will this piece of code work? If not, why?

>>A = [1 2 3; 3 5 6]>>B = [2 4 6; 6 10 18]>>C = B.*A

This code will work because ‘.*’ means multiply eachelement of B with the same element of A. In order touse ‘.*’, A and B need to be the same dimensions.

What does C store?

>>A = [1 2 3; 3 5 6;1 2 1]>>B = [2 4 ; 6 1]>>C = B.*A(2:3,1:2)

c) 4 12 30 6

b) 2 8 18 5

a) 6 20 6 2

d) 10 24 12 1

PLOTTING IN MATLAB

Define x and y and call the plot function

Engineers always add …

Title title(‘y = cos(x)’)

X axis label, complete with units xlabel(‘x-axis’)

Y axis label, complete with units ylabel(‘y-axix’)

Often it is useful to add a grid grid on

Single quotes are used.

Line, Color and Mark Style

You can change the appearance of your plots by selecting user defined line styles mark styles colorTry using

help plotfor a list of available styles

Specify your choices in a string

For exampleplot(x, y, ‘:ok')

strings are identified with single quotes the : means use a dotted line the o means use a circle to mark each

point the letter k indicates that the graph

should be drawn in black (b indicates blue)

Available choicesTable 5. 2 Line, Mark and Color Options

Line Type Indicator Point Type Indicator Color Indicator

solid - point . blue b

dotted : circle o green g

dash-dot -. x-mark x red r

dashed -- plus + cyan c

star * magenta m

square s yellow y

diamond d black k

triangle down v

triangle up ^

triangle left <

triangle right >

pentagram p

hexagram h

specify the drawing parameters for each line after the ordered pairs that define the line

Subplots

The subplot command allows you to subdivide the graphing window into a grid of m rows and n columnssubplot(m,n,p)

rows columns location

subplot(2,2,1)

2 rows

2 columns

1 2

3 4

-20

2-2

02

-5

0

5

x

Peaks

y

2 rows and 1 column

Other Types of 2-D Plots

Polar PlotsLog/semi-log (semilogx,semilogy,etc.)

Bar GraphsPie ChartsHistogramsX-Y graphs with 2 y axesFunction Plots

CREATING FUNCTIONS IN MATLAB

A simple function (poly2)

Save the file as poly2.m (same as the name of the function)

The function is available from the command window or from other M-file programs

Comments

You should comment functions, just as you would any computer codeThe comment lines immediately after the first line are returned when you query the help function

Using find and logical operators in Matlab

Logical Operators

& and~ not| or

find

The find command searches a matrix and identifies which elements in that matrix meet a given criteria.

index numbers

element values

Mathematical manipulations etc…

MATLAB interprets * to mean matrix multiplication. The arrays a and b are not the correct size for matrix multiplication in this example

Multiplication between arrays is performed on corresponding elements if the .* operator is used

Array Operations (element-by-element)

Array multiplication .*Array division ./Array exponentiation .^

In each case the size of the arrays must match


>>A = [1 2 3; 3 5 6]>>B = [2 4 6; 6 10 18]>>C = B./A


1. C=2 2 2

2 2 3

2. C= 2 2 2 2 2 3

3. C=

2 2

2 2

2 3

4. Can’t divide these matrices


>>A = [1 2 3; 3 5 6]>>B = [2 4 6; 6 10 18]>>C = B*A


1. C=2 2 2

2 2 3

2. C= 2 2 2 2 2 3

3. C=

2 2

2 2

2 3

4. Can’t multiply these matrices: not the correct dimension – A is 2x3 B is 2x3

What is A’?

>> A = [1, 2, 3; 4, 3, 6];>> A’

A. 1 4 2 3 3 6

B. 4 1 3 2 6 3

C. 1 2 3 4 3 6

D. 4 3 6 1 2 3

This is the transpose= the rows and columns are flipped


>>A = [1 2 3; 3 5 6]>>B = [1 1 1; 1 1 1]>>C = B*A’


1. C=6 14

6 14

2. C=

3. C=14 14

6 6

4. Can’t multiply these matrices: not the correct dimension – A is 2x3 B is 2x3

2 2 2

2 2 3

What is C?

>> A = [1, 2, 3];>> B = [4, 2, 1];>> C=dot(A,B)

A. 1 4 2 2 3 1

B. 11

C. 1 2 3 4 2 1

D. 4 8 3

Dot product = sum of the elements in vector A multiplied by same elements in vector B

What is C?

>> A = [1, 2, 3; 3, 2, 1];>> B = [1, 1; 1, 1; 1, 1];>> C=A*B

A. 1 4 2 2 3 1

B. 11

C. 6 6 6 6

D. 4 8 3

Matrix multiplication= dot product of each row of A with each column of B

Matrix Inverse

MATLAB offers two approaches The matrix inverse function

inv(A) Raising a matrix to the -1 power

A-1

A matrix times its inverse is the identity matrix

Equivalent approaches to finding the inverse of a matrix

Not all matrices have an inverse

Called Singular Ill-conditioned matrices

Attempting to take the inverse of a singular matrix results in an error statement

Determinants

Related to the matrix inverseIf the determinant is equal to 0, the matrix does not have an inverseThe MATLAB function to find a determinant is det(A)

What is B?

>> A = [1, 2; 3, 2];>> B=det(A)

A. 0 B. 4

C. -4 D. 8

Don’t have to know the mathematics of the determinent

Solutions to Systems of Linear Equations

3 2 10

3 2 5

1

x y z

x y z

x y z

Using Matrix Nomenclature

111

231

123

A

z

y

x

X

1

5

10

B

and

AX=B

We can solve this problem using the matrix inverse approach

This approach is easy to understand, but its not the more efficient computationally

Matrix left division uses Gaussian elimination, which is much more efficient, and less prone to round-off error

What is X?

>> A = [1, 0; 0, 1];>> B = [5; 2]>> X = inv(A)*B

A. 1 1 B. 5 2

C. 5 1 D. 5 2

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Review of MATLAB