Array Operations in MATLAB Video Lecture - MATLAB Programming for Numerical

hello and welcome to the matlab programming
ah nptel course ah we are in module 1 which
is introduction to matlab
lecture 1 point 2 is going to cover arrays
and matrix operations in matlab in the previous
lecture we had given an overview of what matlab
is
in this particular lecture we will learn how
to build arrays which we had started of in
the previous lecture itself
we will go over that again we will talk about
colon notations and we will talk about array
operations and array functions
ah as before there is ah video ah in on the
matlab website that you can watch the link
for which is provided over here
ah the va video is called working with arrays
in matlab and ahm i suggest that you also
look at this this video ah in addition to
this hm nptel lecture ok
so let us look at
building of matlab arrays and vectors ok
ah in the previous lecture we had hm look
at how to build ah aah row vector if we have
ah row seperated eh elements of the row separated
by commas we are going to have a row vector
ah and if instead we were to separated by
semicolon we are going to have a column vector
as shown over here ok
so i am getting an error the reason why i
get an error is because i forgot to put a
semicolon over here
so it is trying to build a vector by having
a one in the 1st row and in the 2nd row we
have 2 elements and we cannot build a vector
of that size so instead if we were to have
semicolons as shown over here we are going
to get a coloumn vector as seen over here
that leads me to ne next thing and that is
that we we do not have to build vectors and
a arrays by having each and individual elements
we can use previously defined ah ah ah vectors
and arrays to build a particular matrix also
so let us say we were to define c as a part
as let let say identity matrix of size 3 ok
so we will get c as eye 3 eye is ah va command
in matlab that will build an identity matrix
ok
so what we have over here is aah a vector
a that is a 3 by 1 so ah sorry ah 1 by 3 vector
and a vector c ah or a matrix c that is a
3 by 3
so if we ss write say d equal to a semicolon
c this is the result that we are going to
get
so the 1st row is what our ahh wah vector
a was and next 3 rows are going to be wht
our aaahh c the array c was
so this is how we have built ah the our matrix
d let us say we try to do this a comma c what
is going to happen is we are trying to build
ahr aah a matrix which by combining an array
which has 1 row with a matrix that has 3 rows
which is going to result
the same similar type of error as we had seen
earlier this is an error in horizontal catenation
that means we are combining ah 2 arrays horizontally
and there is an error in combining that
aah re recall earlier that we had what is
known as a vertical catenation error however
instead if wav to we were to give e equal
to b comma c that is going to work because
this is our ah vector b which has 3 rows and
we are horizontally catenating it with another
vector that has ah another matrix sorry that
has 3 rows and we will be able to do that
the next function is transpose of ah a vector
or a matrix and for transpose you have to
use apostrophe so or dash so if you we say
a dash we are going to get instead of a column
vector ah instead of a row vector we are going
to get a column vector
and just as we did before we we can we can
construct a matrix now saying a dash comma
c now a dash what is going to happen is 1st
we are going to have transpose so 1st matlab
is going to calculate the transpose of a a
recall was a 1 by 3 array so if we transpose
the 1 by 3 array we are going to get a 3 by
1 array so now 3 by 1 array can be horizontally
catenated with a 3 by 3 array why because
we have the same number of rows in a prime
as well as in c so if we are giving this particular
command as you will see later
we will see that we are able to construct
the matrix f so f equal to a dash comma c
enclose in square brackets uh press enter
and we are going to get our matrix f ok now
let us say we wanted to get any particular
element of this matrix so let us say we wanted
to get this particular guide to over here
this is row number 2 and column number 1 so
we can get this by saying f ss squa ah not
square but round brackets f 2 comma 1 so that
is row number 2 column 1 and if we press enter
we are going to get this particular element
likewise let us say we wanted to get this
element that is now in f it is in row number
3 and column number 2 and press enter and
we are sorry row number 2 column number 3
row number 2 and column number 3 and we are
going to get this particular element ok so
when we give this command it is row number
2 the 1st guy is the row and the 3rd column
over here that is column number 3 when we
give this command that is the row number that
is going to be a row number ah 3 and column
number 2 ok so that is what we are going to
get
now if we were to give let us say f 5 comma
5 now as you know that f 5 comma 5 does not
exist because we do not have 5th row nor we
have 5th column if we give this we get an
error it says index exceeds matrix dimensions
so that is because that the ah size of the
matrix was 3 by 4 and our we were trying to
query f 5 comma 5 that does not ex ah exist
ok
so i will clear the screen over here now if
wah i wanted to find the size of that matrix
there is a very simple command called size
if i give size f it is going to return to
me the size ah ok the number of rows is 3
and number of columns is 4
ok
so that is about building the arrays ah i
will next talk about col colon notations so
colon notations is so colon in matlab is nothing
but 2 or say whe want to go from say 1 to
1 to 5 we use a colon 1 colon 5 and let us
look that uh here
if i give 1 colon 5 i have 1 2 3 4 and 5 note
that when we use colon notations matlab creates
a row vector and not a column vector we can
always assign this to any other variable so
let us say we were to say z equal to 1 to
5 this is what we are going to get
now let us say we wanted to have 1 to 5 in
steps of 3 we will give this as y equal to
1 colon 3 colon 5 ok
so in steps of 3 means the 1st guy is 1 2nd
guy is 1 plus 3 that is 4 and the 3rd guy
would be 1 plus 4 plus 3 that is 7 however
7 goes beyond the value 5 so 7 will not be
admissible over here so if i press enter i
will get 1 and 4 as the 2 elements uor ah
in in this particular vector
now if i were to do y equal to let us say
2 colon 1 what is going to happen ok now 2
colon 1 means we want to go from 2 to 1 in
steps of 1 the 1st guy itself 2 is greater
than 1 and therefore we are not going to get
anything we are going to get a blank matrix
ok so y becomes an empty matrix which has
1 row and 0 columns note that matlab is not
giving an error matlab is giving an empty
matrix ok
likewise if you were to du have y equal to
empty square brackets we are going to get
an empty matrix so that is the 0 by 0 matrix
ahh an empty matrix ok
so ah em empty matrices are also matrices
in matlab of size 0
keep that in mind because that is an important
aah ah information ok
so we have kind of covered what colon notations
are ah just as we had seen earlier we can
query any part of of of and ah of a vector
or an array using huh ah elements of that
array likewise we can wah query
mutiple parts of that array so let us look
at the array f over here and let us say we
wanted to get 2nd row and column number 2
and 3 how we can do that is f row number 2
and column number 2 and 3 that is 2 colon
3 and we will be able to get 0 and 1 over
here likewise if we wanted row number 2 and
3 in column 2 we will write f row number 2
to 3 and column number 2 so 2 colon 3 means
row numbers 2 to 3 and column number 2 press
enter this is what we get that is 0 and 0
ok let me clear thisok
in order to query any of the elements or any
of the sub matrix the ah ah ve command that
we are going to use is f p comma q where p
represents all the elements that we want to
query and ah in ah row and q all the elements
that we want query in a column so if you were
to say 3 comma 3 that going to f 3 comma 3
which is 0 likewise say p equal to 0 sorry
p equal to 3 q equal to 3 and if you were
to query f p comma q we are also going to
get f p comma q ok now instead if you were
to say p is 1 comma 2 and if you say f is
p comma q what is that going to do is we are
going to query the 1st and 2nd rows of f and
the 1st column of f because p is 1 comma 2
and q is 3 ok and this is what we are going
to get so f what we queried is because p was
1 comma 2 so 1st and 2nd row q was equal to
3 so column number 3 so 1st and 2nd row of
column number 3 is 1st and 2nd row of column
number 3 is what we get ok
likewise we can say f 1 to 2 and 3 to 4 and
this is what we will get so again let us type
f over here so what we have is f rows number
1 and 2 and columns number 3 and 4 so we are
going to get this particular sub matrix as
our a and s
so what we have now covered so far is how
to build an array and how to use colon notations
both to build an array as well as to get ah
uh parts of an array or sub vectors in an
array or a sub matrix or the sub array in
that particular array ok
this are going to be extremely powerful tools
that we are going to use and leverage in the
rest of this particular course and aah i encourage
you to look at the videos that are there on
the matlab website as as well as well as try
these things out yourself at the end of this
particular module we will wah have a certain
set of examples that ah ah you can try ah
these examples will be split based on the
various lectures and i would encourage you
to start going through this particular examples
right away in parallel to doing this ah video
lecture ok
so let us go to the next slide : ok
huh this is what weah we have use in order
to build the arrays huh we have we uk we use
the comma and semicolon in order to build
an array using numbers as we are shoch shown
over here in addition to that we can also
use existing arrays if there are of an of
a correct size in order to build arrays as
well ok in addition we have also see in this
function eye that is to create i n by n identity
matrix in addition to that there are several
other array building functions such as a 1s
0s diag ha array and d rand and magic hilbert
and so on and so far hhuh
so i will quickly show you how to build hum
aahh how use 1s if we give the command 1s
2 comma 4 we are going to get an array containing
all 1s of of 2 rows and 4 columns this is
what we see over here and that will be assign
to a variable a and s ok
if i give the command say x capital x equal
to 0s 2 comma 1 we are going to get 2 rows
and a single column with the el all of the
elements getting the value 0 so that is what
we matrix functions are going to do
so let us look at another function called
rand rand is basically uniformly distributed
random numbers
so if i give rand without any arguments i
am going to a single number if i give rand
with 2 arguments 2 comma 3 i am going to get
a 2 by 3 matrix of random numbers these are
the uniformly distributed random numbers that
means you will get values between 0 and 1
with equal probability ok
so these are the various array building huh
faah functions
we also saw in the previous lecture some of
the basic mathematical expressions the a scalar
expressions where a logarithm exponent power
ah sin cos and so on there are matrix operations
plus minus multiplication division these are
not ss scalar operations but matrix oprations
so what i am mean by that is that we need
our 2 vectors to be commutated so let us bring
out our vector a ok if i do a star a we are
going to get an error that is because ah 1
by 3 matrix cannot be multiplied by a 1 by
3 matrix the number of columns of a has to
be equal to the number of rows of the 1 that
you are multiplying that a with
so if you give we get an error saying that
inner matrix dimension must agree the res
reason why that happens is a is a 1 by 3 huh
vector
let us say we have a vector b now b was a
3 by 1 vector so we can indeed multiply a
1 by 3 vector with a 3 by 1 vector so you
can indeed do a star b and that is going to
be a matrix multiplication ok
let us look at the vector the matrix e e was
a 3 by 4 matrix where as a is auw 1 by 3 matrix
so indeed we can multiply a with e so if we
do a multiply by e we are going to get ah
1 by 4 vector ok and this is what we get when
we do a multiplied by e again the rules of
matrix multiplications are indeed available
ah or applicable over here ok
the slash we will not go over this slash currently
ah that is because a little bit more advance
at this stage this slash is going to be either
depending on what slash you use either it
is going to be ah left division or a right
division aah what left division and right
division means something that we will go in
1 of the later modules we are not going to
cover that in this particular ma module
aah if you are going to use a slash aah use
that with care and wait until the next module
ul ah ah that is going to cover the slash
operation
the next 1 is a power operation
and that operation is the caret sign the caret
sign is again
aahf ah a matrix ah operand so we cannot use
that caret for a non square matrix that has
to be use on a square matrix itself so if
we do a caret 2 it is same as a star a these
are the same commands and as we had seen a
is not commutative towards self under multiplication
however instead if we had a 3 by 3 matrix
let us say ah ah let us say r equal to rand
3 comma 3 ok that would be commutative in
aah under ah power as well so r to the power
2 is just going to be r star r so its it is
a matrix multiplication and this is what we
will get ok
what i am showing over here also are the scalar
operations and equivalent vector operations
so here if we do exp of ah ss ah of ah aah
an an array we are going to get element by
element exponential if we want a matrix exponential
we want to do expm
ok let us look at that let us say hah if we
had say a matrix d equal to 1 0 2 1 and if
we were to do let me just clear this clc ok
1 0 2 1 if was to do exp of d am to i am going
to get each element going is going to get
exp aah its exponent if instead if i were
to do exp m of d this is what i to get going
to get and this is a matrix exponent and matrix
exponent has its own rules aah something that
we are not going to go over ah in this particular
maa module but this is an individual ah element
by element exponent and this is a matrix exponent
and as you can see that the matrix exponent
is different than element by element exponent
ok in addition to that there are other commands
sum prod cumsum and cumprod let us look at
those commands also
so our matrix a was 1 2 and 5 sum is going
to be just sum of the elements of that particular
matrix so 1 plus 2 plus 5
cumulative sum is cumsum here if you give
cumsum of a what we are going to get is 1st
element is going to itself the 2nd element
is going to be 1 plus 2 the 3rd element is
going to be 2 plus 5 ok
so we get 1 3 and 8 as the result of cumsum
likewise we have prod prod is just multiplication
of all the elements of a and cumprod is aah
multiplication of all the elements until that
particular point so cumprod of a is 1 then
1 multiply by 2 and then 1 multiply by 2 multiply
by 5 and this is a result that we get
ok aah ss this command size we have already
seen length is a length of a particular aah
array so if we have an array which is say
5 by 1 and if we gve command length it will
return the value of 5 ok again these are things
that you should try out in matlab ok now in
addition to the matrix operands we also have
an element by element operand ah recall the
difference between log and logm if we wanted
to do this with multiplication division and
power sign we have an operand called dot star
dot slash and dot caret which will operate
on each element
and let us look at that
let us define our variable a as 1 2 3 1 ok
if i do a star a we are going to get a which
is going to be multiplied by itself ah that
is going to be matrix multiplication however
if i am going to do a dot star a it going
to be each element of a multiplied by itself
so we are going to get 1 multiplied by 1 that
is 1 2 multiplied by 2 that is 4 9 and 1 this
is what we will see when we do the result
over here ok
exact same thing is what you can expect with
a caret 2 ok and that is going to be same
as a star a squaring of a ok and likewise
if you want this similar to to a dot star
a that is going be a dot caret so a dot caret
2 is going to give us 1 4 9 1 that means each
individual element of ah the array has been
squared ok
so these are the basic mathematical expressions
again for people starting out with matlab
some of these operations are ah little bit
difficult to grabs so aah i would strongly
suggest that you go and try out these ah ah
operations ah take a pen and paper along with
you and see what the matrices that you are
forming and what results you are going to
get with that i come to the end of this particular
lecture and i will see you in lecture 1 dash
3 thank you