OPERATIONS

• Advanced matrix operations fall under the
following categories
- Building larger matrices
- Relational operations
- Logical operators and functions
- Subscripting
- Manipulating matrices
- Reshaping

BUILDING LARGER MATRICES

• We can form larger matrices from smaller
matrices.

• Let a=[1 2 3;4 5 6;7 8 9] and b=[4 5 6] then

c=[a ,b’]

adds a 4th column to a.

MATRIX MULTIPLICATION

• MATLAB multiplies two matrices provided
rows and column numbers agree.

• For example, if

A=[1 2 3;4 5 6;7 8 9]
B=[7 8 9;4 5 6;1 2 3]

• Multiplication of A and B is simply A *B

18 24 30
54 69 84
90 114 138

RELATIONAL OPERATIONS

• Relational operators are defined as follows
< less than
<= less than or equal
> greater than
>= greater than or equal
= =equal
~= not equal

WHAT DO THEY DO?

• The response of a relational operator is
another vector

• This vector is binary (0,1) and of the same
length as the data vector

• The 1 locations indicate places where the
relational operator is TRUE

EXAMPLE USAGE

• Define the following row vector

v=[1 3 4 9 8 6 7 0 2]

• Do the following and look at the results.

x=v<5→ 1 1 1 0 0 0 0 1 1
x=v<=4 →1 1 1 0 0 0 0 1 1
x=v>3 →0 0 1 1 1 1 1 0 0
x=v>=0→ 1 1 1 1 1 1 1 1 1
x=v==6 →0 0 0 0 0 1 0 0 0
x=v~=6→ 1 1 1 1 1 0 1 1 1

EXACT LOCATION
IDENTIFICATION: find

• One of the most powerful operators in
MATLAB is find. It operates on a
vector/matrix and returns the positions of
nonzero entries

• find (v>=5) gives the exact locations where
the elements of v equal or exceed 5

• For v=[1 3 4 9 8 6 7 0 2 6]
- v>=5 ---- > 0 0 0 1 1 1 1 0 0 1
- find (v>=5) ------ >4 5 6 7 10

sum function

• The sum function is particularly powerful.

• If x is a vector, sum (x) is simply the sum of
the elements of x

- x=[1 3 4 2 6]
- sum (x)=16

• For a 2D array, sum (A) adds up each column

Combining sum and find

• Find how many times your data , stored in x,
exceeds a threshold
- x=[1 4 3 2 5 7 4 8 9 5 7];%data
- v=find (x>5);%{0,1} pointer array
- n=sum (v);%n equals number of 1's, i.e.
number of times x has exceeded 5

LOGICAL FUNCTIONS

• MATLAB contains a set of logical functions:

• any (x).....returns a 1 if any element in x is
nonzero

• all (x)....returns a 1 if all element in x are
nonzero

• find (x)..returns the indices of nonzero
elements of x

LOGICAL OPERATORS

• Logical expressions can be compared using
logical
operators. There are 3 logical
operators:

not...~
and...&
or......|

• Define a=[1 9 8], b=[2 9 7] and c=[2 5 4 ]. Then
a>b=[0 0 1]
a>c=[0 1 1]

• Then a>b&a>c=[0 0 1].

• You can pick out the elements of an array A
using another array

• Define
- A=[1 2 3;4 5 6;7 8 9],
- v=[1 3]

• Then A(:, v) is another matrix consisting of all
the rows of A but only columns 1 and 3.

• Try A (v,:)

POSITIONS IN A MATRIX

• By using a logical array (0,1) we can point to
specific positions in a matrix.

• For example, let a=[1 3 4 9 8 7 7 0 2 8]. Want to
find values below 6
- v=a<6 ->1 1 1 0 0 0 0 1 1 0
- a(v) --> 1 3 4 0 2

A 2D example

• In an image processing application, we may
want to identify intensities above a threshold
in an image

- a=[7 6 3;6 5 2; 3 4 5];%input image
- v=a>5;%pointer to desired locations
- MATLAB responds v =
1 1 0
1 0 0
0 0 0

A(v) - - > 7 6 6

MAKING AN ARRAY OUT OF A
MATRIX

 •Using (:) by itself strings out all the elements of A in a long column vector • An interesting effect is generated by using A(:)=10:18 • A(:)= 7 4 1 8 5 2 9 6 3

EQUATING MATRICES

• It is important that when matrices or arrays
are equated, the number of rows and columns
match on both sides

• For example, if A is 3x3

- A=ones(4) is invalid because the left hand
side is 3x3 but the right hand side is 4x4.

• The correct assignment is
- A=ones(3)

EMPTY MATRICES

• Statement x=[] defines a zero x zero matrix.
This is different from clear x.

• We can use an empty matrix to efficiently
remove rows and columns from a matrix.

• The following removes columns 2 and 3
entirely

MATRIX MANIPULATION

• Here is a list:

- rot90...............rotation
- fliplr................flip matrix left to right
- flipud..............flip matrix up and down
- diag.................extract diagonal
- tril................... lower triangular part
- triu..................upper triangular part
- reshape...........reshape

REARRANGING MATRICES

• It is possible to rearrange , say, a 3x4 matrix
into a 2x6, 1x12, 4x3 etc. as long as number of
elements do not change

B=reshape(A,m,n)
maps A into an m rows, n columns matrix

reshape(A,2,8)

reshape PRACTICE

• Generate an alternating (+/-) 1 array of length
20 using reshape

[1 -1 1 -1 1 -1...1 -1]

• Hint: first create +1’s and -1’s separately, then
interleave them

USEFUL MATLAB FUNCTIONS

• The following functions are quite handy. Try
them on a random vector of numbers of
length 100.

- max (v)............find the maximum of v
- min (v)............find the minimum of v
- mean (v).........find the mean (average) of v
- std (v).............find the standard deviation
- sort (v)............sort v
- sum (v)...........sum of all the elements in v
- prod (v).......... product of elements in v
- hist (v)............histogram of values

Hands-on exercise

• Do the practices on Page 103 and 99

• The following 2 slides are also part of the
hands-on

Application note

l It is frequently desirable to isolate and
remove noisy spike in data

EXAMPLE: REPLACING AN
OUTLIER

• Want to locate a spike in data and then
remove it
- Plot the data
- Find out what position is the spike located at?
- Remove it and replace it by 0
- Plot your result to see if it has worked

WORKING WITH SOUND

• Let’s try out the previous commands on an
actual sound file.

• In the command window type load bond and
check your workspace to see where the data
went

• You can playback the sound file using sound
command
l Plot the sound file. Where is the data stored
in?

HOMEWORK