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Now that you know how to load and save
data in Octave, put your data into

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matrices and so on. In this video I'd like
to show you how to do computational

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operations on data. And later on we'll,
we'll be these source of computational

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operations to implement our learning
algorithms. Let's get started. Here's my

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active window. Let me just quickly
initialize some variables to use for our

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examples and set A to be a three by two
matrix, and set B to a three by two matrix

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and let's set C to a. Two by two matrix,
like so. Now, let's say I want to multiply

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two of my matrices. So let's say I wanna
compute A times C. I just type A times C.

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So it's a three by two matrix times a two
by2 matrix. This gives me this three by

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two matrix. You can also do elements-wise
operations, and do A. Times B. And what

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this will do, is, it'll take each element
of A, and multiply it by the corresponding

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elements of B. So that's A times B, that's
A. Times B. So, for example, the first

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el-, first element. Is one times eleven
which gives eleven the second element

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gives two times twelve which gives 24 and
so on, so the element wise multiplication

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of two matrices and in general the P rate
tends to is usually used to denote element

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wise operations in octave so here's a
matrix A and if I do A thought to carry

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two this gives me the multi-. The element
wise squaring of A so you know one squared

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is one, two squared is four, and so on.
Let's set V to a vector. We'll set v as

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1,2,3 as a column vector. You can also do
one dot over V to do the element-wise

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reciprocal of V. So this gives me one over
one, one over two, and one over three.

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This [inaudible] matrices. So one dot over
V gives me the reciprocal of one over A.

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And once again the period here gives us a
clue that this is an elements wise

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operation. It also do things like log b
this is a element wise logarithm. The

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element of the b, e to the b is base e
exponentiation of these elements. So this

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is e, this is e squared [inaudible], this
is b. And I can also do abs b to take the

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element wise absolute value of b. So here
you know, b was all positive abs -one to

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say -three. The element wise absent value
get's me back these non-negative values

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and. Negative E gives me the minus of E
this is the same as negative one times B

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but usually just for a negative B so a
negative one times B and What else can you

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do? Here's another neat trick. So, let's
see. Let's say I want to take v and

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increment each other's elements by one.
Well, one way to do it is by constructing

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a 3-by-1. Vector is all 1's and adding
that to D so they do that this increments

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D by from one two three to two three four
the way I did that was length. Of V is V

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so 1's macro V by one this is let one V by
one so let's one V by one. Alright and

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what I did was v plus one three by one,
which is adding this vector of all ones to

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v. So this increments v by one. And you,
another simpler way to do that is to

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actually type v plus one such as v and v
plus one also is to add one otherwise to

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each of my elements of v. Now. Let's talk
about, more operations. So here's my

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matrix A. If you want to write A
transpose, the way to do that is to write

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A prime. That's the, apostrophe symbol, is
the left quote. So in, in, on your

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keyboard, you probably have a left quote,
and a right quote. So this is, a, a,

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excuse me, this is actually a standard
quotation mark, 'cause, you just type A

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transpose. This gives me the, you know,
transpose of my matrix A. And of course, a

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transpose, if I transpose that again, then
I should get back my matrix A. Some more

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useful functions let's say lower case A is
11520.5. So it's a you know one-fourth

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matrix. Let's VAL = max of A this returns
the maximum value of A which in this case

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is fifteen and I can do VAL end max A and
this returns VAL and end which are going

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to be the maximum value of A which is
fifteen as well as the index so is the

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element number two of A that is fifteen so
end is my index into this just as a

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warning if you do max A. Where A is a
matrix. What this does is this actually

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does a column wise maximum. But say a
little bit more about this in a second. So

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using this example, the variable lowercase
a, if I do a less than three, this does

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the element-wise operation, element-wise
comparison, so, the first element of a is

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less than three, [inaudible] it's a one.
The second element of A is not less than

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three, so this evaluates as zero, cuz it's
false. The third and fourth element of a.

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Are of this I mean less than three
[inaudible] less than three so it's just

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one, one. So this does the element wise
comparison of all four elements of

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[inaudible] lower case a to three and it
returns true or false depend on, on

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whether or not it's less than three. Now
if I can find a less than three this will

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tell me which are the elements of A.
Therefore [inaudible] in this case the

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first third and fourth elements are less
than three. For my next example let me set

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A to be equal to magic three the magic
function returns let's type help magic,

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the magic function returns functions
called magic, returns functions called

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matrices squared, they have this your
mathematical property that all of the rows

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and columns and diagonals sum up to the
same thing so you know, is not actually

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useful for machine learning as far as I
know but I'm, I'm just using this as a

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convenient way. You know, to generate a
three by three matrix. And, and these

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magic squares had prop-, had the property
that each row, each column, and the

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diagonals all add up to the same thing. So
it's kind of a mathematical construct. I

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use magic, I use this magic function only
when I'm doing demos, or when I'm teaching

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Octave like this. And I don't actually use
it for any you know, useful machines or

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any application. But let's see if I type
RC=Find A greater than=7, this finds All

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the elements of a that I grid to equals
seven. And so, ASI sets our row in

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columns. So the one, one element is going
to seven, the three two elements go into

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seven, and the two three elements is going
into seven. So, let's see. The two three

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element, for example, is a, two, three is
seven, is this elements. Out here and that

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is indeed greater than seven. By the way I
actually don't even memorize myself what

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these fine functions do and what all of
these things do all myself. And when I use

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the fine function sometimes I forget
myself exactly what it does and you know,

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I type help find to look up. Okay, just
two more [inaudible] I'll briefly show

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you. One is the sum function. So here's my
A and type sum A. This adds up all the

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elements of A and to multiply them
together I take prod A [inaudible] and

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this returns the product of these four
elements of A. [inaudible] these elements

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of a. So 0.5 gets rounded down to zero and
ceiling A gets rounded up, 0.5 rounded up

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to nearest integer, so 0.5 gets rounded up
to one. You can also let's see let me type

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rand V this generates a 3x3 matrix if I
type max rand three rand three what this

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does is it takes the element of Y maximum
of two random 3x3 matrices. So you notice

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all of these numbers tend to be a bit on
the large size because each of these is

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actually the max of a randomly of an
element wise max of two randomly generated

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matrices. This is this is my matrix number
this is my matrix square 3x3. A, let's say

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I type max A Then this with. Open closed
square brackets comma one. What this does

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is [inaudible] column wise maximum. So
maximum of first column is eight, maximum

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of second column is nine, the maximum of
third column is seven. This one

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[inaudible] max along the first dimension
of A. In contrast if I were to take max A

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this plenty notation two then this takes
the per row maximum so the max of the

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first row is eight max of second row is
seven max of the third row is nine and so

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this allows you to take max's either per
row or per column. And if you want,

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remember it defaults to column [inaudible]
wise elements. If you want to find the

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maximum element in the entire matrix A,
you can type max of max of A, like so,

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which is nine, or you can turn A into a
vector and type max of A colon, like so,

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and this treats this as a vector and takes
the max elements of our vector. Finally.

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Let's set A to be a nine by nine magic
square. So remember the magic square has

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this property that every column and every
row sums the same thing and also the

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diagonal. So here is a nine by nine matri
magic square. So let me do sum A one so

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this does a per column sum. So I want to
take each column of A and add them up. And

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this, you know, lesson's verified that
indeed for a nine by nine magic square

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every column adds up to 369, adds up to
the same thing. Now lets do the row wide

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sum so the sum A [inaudible]. Two, and
this. Sums up each row of A and indeed

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each row of A also sums up 2639. Now lets
sum the diagonal elements of A and make

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sure that they that, that also sums up to
the same thing so what I'm going to do is.

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Constructing 9x9 identity matrix that's
I9. And gonna take A and construct you

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know multiply A elements wise, so here's
my matrix A. And to A times I9 and what

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this will do is take the element wise
product of these two matrices and so this

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should wipe out everything in A except for
the diagonal entries and now I'm going to

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do sum, sum A of that and this gives me
the Some of this these [inaudible]

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elements and indeed the 3,6,9 you can sum
up the other diagonal as well so it's top

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left to bottom right you can sum up the
opposite diagonal from bottom left to top

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right the sum of the command for this is
somewhat more cryptic you don't really

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need to know this I am just showing you
this in case any of you are curious but

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let's see. So if you so if you [inaudible]
stands up down but if you do that, that

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turns out to sum up the elements in the
opposite of the other. Diagonal feed that

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also sums up to [inaudible] here I'll show
you where as I9 is this matrix, that.

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Uptown of I9. You know takes the identity
matrix and flips it vertically so that you

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end up with excuse me [inaudible] and that
with 1's on this opposite diagonal as

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well. Just one last command and then
that's it and then that'll be it for

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[inaudible]. Lesser a to be the magic, the
magic square game and you want to invert a

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matrix you type p is a. This is
[inaudible] called a pseudo inference, but

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it doesn't matter. Just think of it as
basically inverse of a and that's the

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inverse of a. And so you can set, you
know, temp equals pn of a and of a temp

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times a. This is indeed the identity
matrix where essentially ones on the

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diagonal and zeros on the off diagonal up
to a numerical round off. So, that's it

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for how to do, different computational
operations on the data and matrices. And,

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after running a learning algorithm, often,
one of the most useful things is to be

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able to look at your results. So it's a
plot, a visualizer result. And in the next

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video, I'm going to very quickly show you
how, again, with one or two lines of code,

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using [inaudible], you can quickly
visualize your data or plot your data.

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And, use that to better understand, you
know, what your learning algorithms are