1. Lecture 2 MATLAB as Calculator Scalar Variables Vectors and Matrices
Basic Matrix Operations

2. Useful Functions and Operations MATLAB
MATLAB as Calculator (Revisited)
>>
2+2
100-43
3*3
5/2
2^8
log(2)*sin(pi/2)
acos(-1)/exp(4)
MATLAB prompt

3. Arithmetic Operations
MATLAB Command
+ (addition) +
- (subtraction) -
× (multiplication) *
/ (division) /
(exponentiation)
^ (caret)

4. Special Values: Normally, mathematicians use i to represent
Description
MATLAB Command
Example Pi pi
x=pi^2
= square root of -1
i or j
i=sqrt(-1)
infinity
Inf
z=pi/0
The last computed unassigned result to an expression typed in the Command window.
ans
Normally, mathematicians use i to represent
whereas engineers use j to represent

5. Trigonometric Functions
Description
MATLAB Command
Example
sine
sin
sin(pi/6)
cosine
cos
cos(pi/3)
tangent
tan
tan(pi/4)
inverse sine (arcsine)
asin
asin(1/2)
inverse cosine (arccos)
acos
acos(1/2)
inverse tangent (arctan)
atan
atan(1)

6. Hyperbolic Functions Description MATLAB Command Example
hyperbolic sine
sinh
sinh(pi/6)
hyperbolic cosine
cosh
cosh(1)
hyperbolic tangent
tanh
tanh(pi/4)
inverse hyperbolic sine
asinh
asinh(0.5)
inverse hyperbolic cosine
acosh
acosh(0.5)
inverse hyperbolic tangent
atanh
atanh(0.5)

7. Exponential, Square Root, Error Functions
Description
MATLAB Command
Example
exponential
exp
exp(1)
natural logarithm
log
log(10)
Common (base 10) logarithm
log10
log10(10)
square root
sqrt
sqrt(25)
error function
erf

8. Complex Numbers: i=sqrt(-1) Description MATLAB Command Example
absolute value (magnitude)
abs
z1=1+j;
y=abs(z1)
phase angle (in radians)
angle
phi=angle(z1)
complex conjugate
conj
z2=4+5*j;
conj(z2)
real(z2)
imag(z2)
complex imaginary part
imag
v=imag(z1)
complex real part
real
z3=2*exp(j*pi/6)
w=real(z3)

9. Rounding Functions
Description
MATLAB Command
Example
round
round(2.5)
round towards negative infinity
floor
floor(-2.6)
round towards positive infinity
ceil
ceil(-2.6)
round towards zero
fix
fix(-1.9)
remainder after division
mod
mod(7,3)
The ceiling of x is the smallest integer which is greater than or equal to x, while the floor of x is the largest integer less than or equal to x.

10. Rounding Functions Description MATLAB Command Example
factorial n or n!
factorial
factorial(4)
binomial coefficient Cnk
nchoosek
nchoosek(4,2)
prime factors
factor
factor(12)

11. Example 1: Question: Compute . In MATLAB this is done by simply typing
at the prompt.
Be careful with parentheses and do not forget to type * whenever you multiply!
>>

12. Useful Functions and Operations MATLAB
Match regular expression (case insensitive):
This means that MATLAB knows a difference between letters written as lower and upper case letters, e.g., a and A.

13. Example 2: >> close all; clear all; clc;
>> balance = 1;
>> Balance
??? Undefined function or variable 'Balance'.
>> BALANCE
??? Undefined function or variable 'BALANCE'.
Upper and lower case characters are not equivalent (MATLAB is case sensitive).
Do you mean:
>> balance

14. Example 3:
>> close all; clear all; clc;
>> balance = 1;
>> balance
balance = 1
Typing the name of a variable will cause MATLAB to display its current value.
MATLAB uses parenthesis ( ), square bracket [ ], and curly braces { }, and most standard characters, such as a to z, A to Z, %, ^, +, -, *, /.

15. Obtaining Help on MATLAB commands
Type help topic to access online help on a command, function or symbol topic (e.g. help cos). Hyperlinks, indicated by underlines, are provided that will take you to related help items and the Help browser.
Use lookfor topic (e.g. lookfor cosine) if you are not certain what the name of the topic is.
If you press the tab key after partially typing a function or variable name, MATLAB will attempt to complete it, offering you a selection of choices if there is more than one possible completion.

16. Scalar Variables in MATLAB
x = 2 x
y=4
x+y
clear x
MATLAB prompt
This assigns the value 2 to the variable x.
Hit return/enter key and what will you see?
Clear a user – definded variable x.
Hit return/enter key and what will you see?

17. Scalar Variables in MATLAB
z = 3;
MATLAB prompt
This assigns the value 3 to the variable z without printing it.
You can have variable name like: X, A2, x_y.
Cases are sensitive: x is not the same as X.

18. Variable identifier (name):
Variable Assignment
>> max_grade = 100;
Assignment operator
Variable identifier (name):
Letters
Digits
Underscore
Can’t start with a number or underscore
Case sensitive!!!
Can’t be a keyword
Value
clear max_grade;
clear;
clear all;

19. Example 4: >> r3d3 = 10; Here are examples of valid variables.
>> pay_cheque = 10000;
Here are examples of valid variables.

20. Example 5: Here are examples of invalid names are:
>> pay-cheque = 10000;
>> 1a = 2;
>> Name$ = 300;
>> _4b = 45;
pay-cheque = 10000; |
Error: The expression to the left of the equals sign is not a valid target for an assignment.
1a = 2; |
Error: Unexpected MATLAB expression.
Name$ = 300; |
Error: The input character is not valid in MATLAB statements or expressions.
_4b = 45; |
Error: The input character is not valid in MATLAB statements or expressions.

21. Example 6: Don’t override built-in functions!!! >>
disp = 100;
disp(x);
builtin('disp', x);
clear disp;
The disp variable overrides the
built-in disp function.
Don’t override built-in functions!!!

22. Built-in Variables Three built-in variables are: pi, i, j, where
Not good to re-assign pi as a variable. To reset i:
>> clear i or
>> i = sqrt(-1)

23. Everything in MATLAB is an array!
Data Types (Variable types)
A scalar is also an array of size 1x1
Next lectures…
Everything in MATLAB is an array!

24. Data Types (Variable types)
Integer
Default

25. Data Types (Variable types)
Integer:
a = int16(100);
Be careful of
memory overflow
b = int8(5000);
Real(/Complex):
x = double(235.5);
x = single(235.5);
x = 235;

26. Numeric Data Functions
A summery of numeric functions:
Help -> contents -> MATLAB -> Functions -> Mathematics -> Array and Matrices (and also the rest of the items…)
Notice:
isinf, isnan, floor, ceil
Infinite
Is Not A Number

27. Who’s Who in My Memory To know what you have assigned or computed:
>> close all; clear all; clc;
b = 2
c = [2,3,5]
d = [4;1;3;-1]
e = 'words, words, words'
f = [7,9;8,5]
who
whos
List variables currently in the workspace
List variables currently in the workspace with their size

28. Vectors and Matrices How to input: rvec = [1 2 3] cvec = [4 3 4 2]
A MATLAB variable is an array.
A one dimensional array is named vector.
A two dimensional array is named matrix.
How to input:
rvec = [1 2 3]
cvec = [4 3 4 2]
mat = [3 3 4
4 5 6
9 9 0]

29. Faster way of generating vectors and matrices:
Or simply:
rvec = [1, 2 3]
cvec = [4; 3; 4; 2]
mat1 = [3 3, 4; 4 5 6; 9, 9 0]
Faster way of generating vectors and matrices:
zeros(4,3) = 4-by-3 zero matrix
ones(1,4) =[1, 1, 1, 1]
eye(4,4) =4-by-4 identity matrix
rand(4,1) =4-by-1 random matrix
[1:4] =[1, 2, 3, 4]
[1:4]’ = ?
[1:0.5:3] =[1, 1.5, 2, 2.5, 3]
[3:-0.5:1] =[3, 2.5, 2, 1.5, 1]
Space / comma – New element in same line
New line / Semicolon – new line
Colon operator – start:increment:end. Default step is one.
‘ operator- transpose (near the enter key)

30. Controlling Output
>> x= ;
>> format long; x
>> format long e; x
e+000
>> y= ;
>> format short e; y
2.3909e+002
>> format short; y
Scientific or
Exponential notation

31. Controlling Output 7 5 6 7 8 3 6 7 8 >> A=[1,2,3,4;5,6,7,8];
>> A(:,3) 3
>>A(2,2:4)

32. Controlling Output >>A=[1 2 3 4; 11 12 13 14; 21 22 23 24] A =
>> A([1 2], [2 3 4])
ans =
>> A([1 3], [1 4])
>> A([1 3], [1 4]) = [-5 -6; -7 -8]

33. Assignment into Array
>> A = [1, 2, 3; 4, 5, 6; 7, 8, 9]
A =
>> A(3,4)
??? Index exceeds matrix dimensions.
>> A(4,4) = 10 1 2 3 4 5 6 7 8 9 ? 1 2 3 4 5 6 7 8 9 10
Notice the difference!

34. Subscripted assignment dimension must match!
Assignment into Array
>> A = [1, 2, 3; 4, 5, 6; 7, 8, 9]
A =
>> A(:) = 21:30
??? In an assignment A(:) = B, the number of elements in A and B must be the same.
>> B = [21, 22, 23; 24, 25, 26]
B =
>> A(1:2,1:3) = B 1 2 3 4 5 6 7 8 9
Notice the difference!
>> A(1:2,1:3) = 10
A = 26 25 24 23 22 21
Subscripted assignment dimension must match!
Scalar expansion

35. Some Useful Commands MATLAB Command Explanatory Note quit, exit
quit MATLAB
diary
log the session
clear
clear all variables in the memory
help XX
help on the command XX
who
what variables in memory
whos
size of the variables
lookfor XY
search command related to XY
clc
clear command window ;
suppress printing %
comments (MATLAB will not read)

36. Entry-wise Operations
MATLAB Command
Explanatory Note
C = [3 1 1; ; ]
% C is a 3-by-3 matrix
d = [2 3 4]
% d is a row vector
e = d'
% e is transpose of d
size(C)
% size of C
length(d)
% length of d
3*C
% scalar multiplication
C*e
% matrix-vector product
C*C
% matrix-matrix product
C\e
% solution to C x = e
d/C
% solution to d = x C
C^2
% square of C = C*C

37. Basic Matrix Operations
MATLAB Command
Explanatory Note
3.*A
= 3*A
A + B
= A+B
A – B
= A-B
A.*B
= ai,j * bi,j
A./B
= ai,j / bi,j
A.^2
= ai,j * ai,j
A+3
= ai,j + 3

38. Mathematical equivalent
Use / and \operations:
MATLAB notation
Mathematical equivalent
Right division:
a/b
Left division
a\b
b/a
Elementary matrix and array operations:
Arithmetic Operation
Matrix sense
Entry-wise sense
Addition +
Subtraction -
Multiplication * .*
Left division \ .\
Right division / ./
Exponentiation ^ .^

39. Scalar Functions for Matrices
MATLAB Command
Explanatory Note
sin(C)
% sine of each entry of matrix C
exp(C)
% exponential of each entry of C
real(C)
% real part of each entry of C
ceil(C)
% round each C’s entry up
floor(C)
% round each C’s entry down
sqrt(C)
% square root of each entry of C
C^(1/2)
% matrix square root of C

40. Scalar Functions as Matrix Functions
Polynomial Evaluation:
Consider p(x)=2x3+4x-7, to compute p(3), use:
p=[ ];
x=3;
y=polyval(p,x)
To compute p(x) for x in between 0 and 2, use:
y=polyval(p,[0:0.01:2])
If A is a matrix, polyval(p,A) gives
2*A.^3+4*A-7
polyval(p,A)