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(What Are Strings?)
A string is an array of characters
Strings have many uses in MATLAB
• Display text output
• Specify formatting for plots
• Input arguments for some functions
• Text input from user or data files
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(Creating Strings)
• We create a string by typing characters within single quotes
(').
 Many programming languages use the quotation mark (")
for strings. Not MATLAB!
• Strings can have letters, digits, symbols, spaces.
 To type single quote in string, use two consecutive single
quotes, e.g., make the string of English "Greg's car" by
typing
'"Greg''s car"'
'ad ef', '3%fr2', '{edcba:21!',
'MATLAB'
 Examples:
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You can assign string to a variable, just like
numbers.
>> name = 'Sting'
name =
Sting
>> police = 'New York''s finest'
police =
New York's finest
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In a string variable
• Numbers are stored as an array
• A one-line string is a row vector
 Number of elements in vector is number of
characters in string
>> name = 'Howard the Duck';
>> size( name )
ans =
1 15
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Strings are indexed the same way as
vectors and matrices
• Can read by index
• Can write by index
• Can delete by index
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Example:
>> word = 'dale';
>> word(1)
ans = d
>> word(1) = 'v'
word = vale
>> word(end) = []
word = val
>> word(end+1:end+3) = 'ley'
word = valley
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MATLAB stores strings with multiple lines
as an array. This means each line must
have the same number of columns
(characters).
>> names = [ 'Greg'; 'John' ]
names =
Greg
John
>> size( names )
ans =
2 4
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Problem
4 characters
3 characters
>> names = [ 'Greg'; 'Jon' ]???
Error using ==> vertcat
CAT arguments dimensions are not
consistent.
Must put in extra characters (usually spaces) by
hand so that all rows have same number of
characters
>> names = [ 'Greg'; 'Jon ' ]
Greg
Extra space
Jon
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(String Padding)
Making sure each line of text has the same number of
characters is a big pain. MATLAB solves problem with char
function, which pads each line on the right with enough spaces
so that all lines have the same number of characters.
>> question=char('Romeo, Romeo,','Wherefore art thou', 'Romeo?')
question =
Romeo, Romeo,
Wherefore art thou
Romeo?
>> size (question)
ans =
3 18
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R o m e o ,
W h e r
e f
R o m e
o ?
R o m e o ,
o r
e
a
r
t
t
h o u
Three lines of text stored in a 3x18 array.
MATLAB makes all rows as long as
longest row.
First and third rows above have enough
space characters added on ends to
make each row 18 characters long.
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Array Operators
• A.* B
multiplies each element in array A times the corresponding element
in array B
• A./B
divides each element in array A by the corresponding element in
array B
• A.^B
raises each element in array A to the power in the corresponding
element of array B
Addition and Subtraction
• Use + to add two arrays or to add a scalar to an array.
• Use – to subtract one array from another or to subtract
a scalar from an array.
 When using two arrays (vectors or matrices), they
must both have the same dimensions (number of
rows and number of columns).
 Vectors must have the same dimensions (rows and
columns), not just the same number of elements
(1x5 is not the same as 5x1).
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Addition and Subtraction
When adding two matrices A and B, MATLAB adds the
corresponding elements, i.e.,
• It adds the element in the first row and first column of
A to the element in the first row and column of B.
• It adds the element in the first row and second column
of A to the element in the first row and second column
of B.
• Etc...
This called elementwise addition.
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Addition and Subtraction
When subtracting two arrays A and
B, MATLAB performs an
elementwise subtraction.
In general, an operation between
two arrays that works on
corresponding elements is called
an elementwise operation.
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Addition and Subtraction
When adding a scalar to an array,
MATLAB adds the scalar to every
element of the array.
When subtracting a scalar from an
array, MATLAB subtracts the scalar
from every element of the array.
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Array Multiplication
There are two ways of multiplying matrices – matrix
multiplication and elementwise multiplication.
MATRIX MULTIPLICATION
• The type used in linear algebra.
• MATLAB denotes this with an asterisk (*).
• The number of columns in the left matrix must be
same as number of rows in the right matrix.
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Matrix Multiplication
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Matrix Multiplication
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>> A = randi(3,3)
A =
3
3
1
3
2
2
1
1
3
>> B=randi(3,3)
B =
3
3
1
1
2
2
3
3
3
>> BA = B*A
BA =
19
16
11
9
21
18
>> AB == BA
ans =
0
0
0
0
0
0
12
11
18
1
0
0
>> AB = A*B
AB =
15
18
12
17
19
13
13
14
12
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Vector Multiplication
When performing matrix multiplication on two vectors:
• They must both be the same size.
• One must be a row vector and the other a column
vector.
• If the row vector is on the left, the product is a scalar.
• If the row vector is on the right, the product is a
square matrix whose side is the same size as the
vectors.
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Vector Multiplication Examples
>> h = [ 2 4 6 ]
h =
2
4
6
>> v = [ -1 0 1 ]'
v =
-1
>> h * v
ans =
4
>> v * h
ans =
-2
0
2
-4
0
4
-6
0
6
0
1
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Scalar (Dot) Product
dot(a,b) computes the
inner (scalar) product.
• a and b must be of the same
size.
• Any combination of vertical
or horizontal vectors.
• Result is always a scalar.
EXAMPLE
>> h = [ 2 4 6 ]
h =
2
4
6
>> v = [ -1 0 1 ]'
v =
-1
0
1
>> dot(h,v)
ans =
4
>> dot(v,h)
ans =
4
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Description
• C = 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 are both 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 B must have the same
size.
• 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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Cross Product
• cross(a,b) computes the
cross product of two vectors. .
It results in a vector which is
perpendicular to both and
therefore normal to the plane
containing them. It has many
applications in mathematics,
physics, and engineering.
• More info:
EXAMPLE
>> a = [ 1 2 0 ]
a =
1 2 0
>> b = [3 4 0 ]
b =
3 4 0
>> cross(a,b)
ans =
0 0 -2
http://en.wikipedia.org/wik
i/Cross_product
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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.
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
d = dot(a,b)
d = 32
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Identity Matrix
A square matrix with ones on main diagonal and zeros
elsewhere.
• When we do a matrix multiplication on any array or
vector with the identity matrix, the array or vector is
unchanged.
 True whether multiply with identity matrix is on the
left or on right
• MATLAB command
eye(n) makes an n×n identity
matrix
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Determinants
A determinant is a function associated with
square matrices
• In math, determinant of A is written as det(A) or
|A|
• In MATLAB, compute determinant of A with
det(A)
• A matrix has an inverse only if it is square and its
determinant is not zero.
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Determinants Example
with Cross Product
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In math, inverse of a matrix A is written as A-1
In MATLAB, get inverse with A^-1 or inv(A)
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Left division, \:
Left division is one of MATLAB's two kinds of array
division
• Used to solve the matrix equation AX=B
 A is a square matrix, X, B are column vectors
 Solution is X = A-1B
In MATLAB, solve by using left division operator (\),
i.e.,
>> X = A \ B
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When solving set of linear equations, use
left division, not inverse, i.e., use X=A\B
not X=inv(A)*B
Left division is
• 2-3 times faster
• Often produces smaller error than inv()
• Sometimes inv()can produce erroneous
results
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Right division, /:
Right division is the other kind of
MATLAB's array division
• Used to solve the matrix equation XC=D
 C is a square matrix, X, D are row vectors
 Solution is X = D·C-1
In MATLAB, solve by using right division
operator (/), i.e.,
>> X = D / C
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Array Division
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Element by Element Operations
Another way of saying elementwise
operations is element-by-element
operations.
• Addition and subtraction of arrays is always
elementwise.
• Multiplication, division, exponentiation of arrays
can be elementwise.
• Both arrays must be same dimension.
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Element by Element Operations
Do elementwise multiplication,
division, exponentiation by putting a
period in front of the arithmetic
operator.
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Element by Element Operations
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Page 73 of text book!
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ELEMENTWISE MULTIPLICATION
• Use .* to get elementwise multiplication (notice period
before asterisk)
• Both matrices must have the same dimensions
>> A = [1 2; 3 4];
>> B = [0 1/2; 1 -1/2];
>> C = A .* B
>> C =
0
1
3 -2
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If matrices not same dimension in elementwise multiplication,
MATLAB gives an error:
>> A = [ 1 2; 3 4];
>> B = [1 0]';
>> A .* B % Meant matrix multiplication!
??? Error using ==> times
Matrix dimensions must agree.
>> A * B % this works
ans =
1
3
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Element by Element Operations
Be careful – when multiplying square
matrices:
• Both types of multiplication always work.
• If you specify the wrong operator, MATLAB
will do the wrong computation and there
will be no error!
 Difficult to find this kind of mistake.
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EXAMPLE
>> A = [1 2; 3 4];
>> B = [0 1/2; 1 -1/2];
>> A .* B
>> ans
0
1
3 -2
>> A * B
ans =
2.0000
-0.5000
4.0000
-0.5000
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Using Arrays in Matlab Built-in Functions
Built-in MATLAB functions can accept arrays as inputs
• When input is array, output is array of same size with each
element being result of function applied to corresponding input
element
 Example: if x is a 7-element row vector, cos(x) is
[cos(x1) cos(x2) cos(x3) cos(x4) cos(x5) cos(x6) cos(x7)]
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BUILT-IN FUNCTIONS FOR ANALYZING ARRAYS
MATLAB has lots of functions for operating
on arrays. For a vector v
• mean(v) – mean (average)
• max(v) – maximum value, optionally with index
of maximum ([C,I] = max(v))
min(v) – minimum value, optionally with index of
minimum ([C,I] = min(v))
• sum(v) – sum
• sort(v) – elements sorted into ascending order
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BUILT-IN FUNCTIONS FOR ANALYZING ARRAYS
• median(v) – median
• std(v) – standard deviation
• dot(v,w) – dot (inner product); v, w both vectors
of same size but any dimension
• cross(v,w) – cross product; v, w must both have
three elements but any dimension
• det(A) – determinant of square matrix A
• inv(A) – inverse of square matrix A
• See Table 3-1 in text book for details
on the preceding functions.
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