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circuits and electronics image
processing computer graphics quantum
mechanics the Google page rank algorithm
any kind of network other stuff these
are the kinds of things where matrices
are used and extremely important for
understanding or analyzing different
systems and although I can't discuss
everything in one video this should give
you some insight into the applications
of matrices beyond an introductory
course might include now in the
beginning matrices can be one of the
most boring subjects we learn in math
maybe now for everyone but least that's
how it was for me I mean we're told hey
here's how matrix addition works real
simply just some of the corresponding
entries and you have your answer
then multiply a matrix by a single
number is as simple as multiplying every
entry by that value but when it comes to
matrix multiplication we do this weird
row by column dot product multiplication
which some teachers just give no context
to so yeah this isn't the kind of stuff
that makes you in a major in matrix math
anytime soon I mean you might learn more
in high school but overall a lot of it
just isn't that exciting however I
promise matrices are used way more than
you probably think but the first thing
we need to realize is that matrices do
things two vectors don't take this as a
definition cuz it's obviously not but we
do need to see what happens when we
multiply a matrix by a vector for
example a vector that starts at the
origin and ends at 1 comma 1 can be
written in matrix form as shown X
component on the top and Y component on
the bottom and when you multiply by a
2x2 matrix like this through the
multiplication rules we get a new vector
out in this case of 1 comma 3 so we put
a vector in and the matrix scaled and
rotate it to get a new vector out this
is what I mean by the matrix doing
things to the vector and in this case
different inputs will be rotated and
scaled differently which we'll see in a
sec now some matrices are much simpler
like this one here just rotates put a
vector in aka multiplied by the matrix
and out will come the same vector
rotated 90 degrees counterclockwise
this matrix on the other hand will just
scale any vector that goes in comes out
twice as long but most two-by-two
matrices like this when we were
analyzing aren't as simple different
input vectors I'll just put a few here
as an example gets scaled and rotated
differently however the transformations
are all linear as in any vector on the
same line as one of those inputs will be
mapped to a vector on the same line as
the corresponding outputs these linear
transformations are why we called the
first in-depth class on matrices linear
algebra anyways I'm going to redo those
transformations once more but this time
pay attention to this vector here you'll
notice it's the only one that is just
scaled it doesn't rotate at all and this
would happen to any vector on that same
line because of what we just saw any
vector that is only scaled by a matrix
is called an eigen vector of that matrix
and how much the vector is scaled by or
two in this case since the length
doubled is known as the eigen value I'm
not going to go through how to solve for
these but the vocabulary will come up
later now the last thing to mention is
that the first application of matrices
we typically learn is how they help us
solve systems of equations the
coefficients can go into a matrix the
variables go into another and the
outputs go here notice I'm using the
same matrix as the one from the last
slide by the way using the rules of
matrix multiplication you can see that
this and this are the exact same so
really what this is asking is which
input vector does this matrix map to 1
comma 3 well we already saw the answer
to this 1 comma 3 is this output and the
question of which vector will the matrix
map to this involves us just doing the
same transformations as before but
backwards to find the answer is 1 comma
1 that will be our solution going
backwards is like applying an inverse
matrix and when the desired output is
the one being multiplied then the vector
it came from comes out so x equals 1 and
y equals 1 are the solutions if we plug
those into the
some equations than both are satisfied
which is exactly what we were looking
for if you haven't seen three blue one
Browns essence of linear algebra series
definitely check that out this will make
a lot more sense but here we're focused
on applications which we're going to get
to now now in systems of equations get
more complex all we have to do is expand
our matrix and we can analyze the system
with as many variables as we want the
reason matrices are used in circuits and
electronics for example is because these
can be represented by linear equations
in which all the voltages and currents
are the unknown variables when the
circuits get hectic where we don't want
to solve it by hand we can just have a
computer find an inverse matrix and
we'll have our currents and voltages but
that's still not too exciting so what
about a system that continuously evolves
over time like for example let's say
there's a zombie outbreak at the local
high school
pretty standard situation and the place
is quarantined so no one can go in or
out but the zombie infection is
spreading so we've got humans in the
school and zombies but no one is coming
in or out so the population remains the
same now let's say every hour 20% of
humans will turn into zombies due to
being infected there is a cure for the
disease luckily however it's not always
guaranteed to work so we'll say that
every hour 10% of the zombies will
return back to humans at this moment if
there are 150 zombies and 150 humans the
question is what is going to happen in
the long run now we're going to assume
the changes happen in discrete intervals
at the hour so let's see what happens in
the first hour for the humans of the 150
starting out 80% of them are going to
stay human or not become infected but we
also have to add the 10% of the 150
zombies that become cured and turned
back to human
this leaves us with a hundred and
thirty-five humans after that first hour
it went down just a bit for the new
total of zombies we would take the 20%
of humans that got infected plus the 90%
of the 150 zombies that are not cured
giving us a total of of course 165 since
these two numbers add together muster
300 but we want to know what happens
after a long time so we got to keep
going after another hour we write the
same percentages except now the number
of humans is 135 instead of 150 and for
the zombies we got 165 instead of 150
this now puts 125 for the humans and 175
for the zombies so we seem to keep
losing humans but will this continue
well what we have here is some linear
equations that can be represented as a
matrix of those percentages which don't
change this is multiplied by the inputs
hmz or the current population of humans
and zombies at any time and all of this
equals the populations after that given
our this is called a markov matrix by
the way since its column sum to 1 and it
has no negative values but this is like
what we just saw the matrix we have is
going to do stuff to or scale and rotate
the input vector the first input was 150
comma 150 the initial human and zombie
populations and after 1 hour or
multiplication it gets moved to 135
comma 165 but we have to keep going and
apply another transformation sending it
to 125 comma 175 the populations we
found after 2 hours so as we keep
applying these multiplications the real
question is where does this vector go
well let me put a few vectors on the
graph to show this each representing
populations which add to 300 if we do
the matrix multiplication and look at
the transformations you'll notice this
vector or anything on this line stays
put while everything else moves towards
it that vector is an eigenvector of our
matrix the associated eigenvalue is 1
since it doesn't scale and since that
vector doesn't rotate or scale it is the
equilibrium of the system and therefore
the answer to our question after a long
time the populations will settle to
numbers which lie on this line and add
to 300 which would be 100 for the humans
and 200 for the zombies any other
population values we'll just move a
little closer to these after each hour
if you put those values inside the
equations from before you'll see the
output remains 100 and 200 for the
humans and zombies respectively then if
the percentages were to change the
question just comes down to what is the
eigenvector of the new matrix there may
not be a zombie infestation anytime soon
but this kind of math could be used to
analyze how a virus will spread
throughout a population for example and
one of my favorite applications of this
is the Google page rank algorithm which
involves Markov matrices and ranks
websites by treating outgoing links as
probabilities of transitioning from one
site to another for more on that I have
a dedicated video which I'll link below
now moving on here's a happy story not
at all on April 29 1992 a man named
Reginald Denny was beaten nearly to
death live on national TV and this was
just a completely innocent man who had
done nothing wrong you can see Reginald
laying here probably unconscious after
the attack the attack itself can be seen
here on YouTube but in an attempt to
knock an age-restricted like that video
is I'm only showing the portion right
after now the back story here is that
April 29th 92 was the first day of the
Rodney King riots in Los Angeles
Reginald Denny was a truck driver whose
route for the day involved going through
an area where rioting was taking place
which he was not informed about when he
got there he was stopped by rioters
dragged out of his truck and that's when
the beating took place now identifying
who assaulted Denny was not easy since
the quality of the live footage wasn't
amazing but what help law enforcement
confidently identify one of the
attackers with some advanced math to
understand how this was accomplished we
need to first look at what a digital
images a digital image when you look
real closely is just made up of a bunch
of pixels each of a single color those
colors can then be represented by some
numerical value which means like a
square picture made up of a million
pixels 1000 on each edge could be
represented by a thousand by 1,000
matrix where the entries are the color
values of each pixel
working with black-and-white pictures is
much easier though because the black
pixel come you represented with a zero
and a white can be a 1 let's say well
actually work with grayscale images here
though meaning anything in between zero
and one can exist which will correspond
to a different shade of grey so when it
comes to image processing and
manipulation whether it be blurring an
image detecting edges sharpening an
image and so on it all comes down to
manipulating the pixels in a very
specific way to see an example let's
mathematically blur this image of the
number 1 to do so I'm going to make a 3
by 3 matrix where every entry is 1/9
this is known as a kernel and image
processing by the way then what we're
going to do is lay this kernel over our
image matrix and multiply the individual
entries in each square together then add
the results in this case it's just 1
times 1/9 nine times so the sum of all
those is one yes this kernel is really
just finding the average of the pixels
inside it from there we're going to take
that sum of one and set the center pixel
to that color in the new image it just
so happened not to change it's still 1
or white but that won't always be the
case now you'll notice this grid on the
right where the blurred image will go is
the same size as the one on the left to
get the entire blurred image we're just
going to sweep that red section across
the original one when we slide it over
once all those pixels are still 1 so the
average is also 1 and that's where this
new pixel becomes but after sliding over
again the kernel contains a black pixel
so we find the average of 8 ones and a
zero which is about point eight nine
this corresponds to a very light shade
of gray which we will put in that middle
square after sweeping across the image
mapping each new number to the blurred
image these will be the values I know
this method doesn't really account for
the border but for our purposes we're
just going to keep that white now I'll
actually color and the pixels based on
their values and we get a blurred image
of the number one
actually this is extremely blurred
almost beyond recognition but if we put
an outline along the colored region we
can see the one is still kind of there
the reason this is so blurred is because
we're only working with a hundred pixels
but what the kernel did was kind of took
this sharp edge in the original image
between black and white
and smooth or averaged it so we get this
fading from dark to light in the blurred
image the kernel we use represents a
type of blur known as a box blur and
from Wikipedia if you input a picture
with many more pixels then apply the
blur this is the output you get but
there are several other kernels that'll
all accomplish different things a
Gaussian blur also of course blurs the
image but it assigns more weight to the
middle square so dark pixels stay fairly
dark and vice versa there's a sharpen
kernel and there's edge detection
kernels which search for sharp changes
in color you'll notice that all the
numbers in this kernel sum to zero so if
we put it over a section of an image
where all colors are roughly the same
multiplying by these numbers then adding
the results would just yield zero or a
black pixel which is why that
corresponding area is black the only
sections that aren't are where we find
sharp changes in color aka an edge as
another example of edge detection here's
a poorly taken photograph of someone's
arm and by using edge detection
algorithms researchers were able to
identify a region of some kind of
birthmark or tattoo well this is
actually a zoomed in portion of this
image where those men can be seen
beating Reginald any using image
processing techniques similar to what
we've seen one company was able to
determine that this mark was a rose
tattoo affiliated with a certain gang in
Los Angeles and it was this that helped
him eventually secure a conviction of
one of the perpetrators of the attack
now all this may not have involved much
matrix math like we saw earlier but no I
did simplify some things to avoid going
in too much detail and not only image
processing but computer graphics heavily
use matrices with these what we can do
is take geometric data and incorporate
it into a coordinate system we can then
scale rotate reflect shift images and
more through matrix manipulation but
things do get much more complicated like
when you want to project a 3d image into
a 2d plane we can use matrix map to map
the 3d points and find where they would
appear on the flat screen not going to
go into much more detail in that but
again computer graphics are another very
useful application of matrices but for
those wanting some real tangible results
that come from matrix math let's look at
networks and graph theory graphs can
represent a lot of things people and who
they're friends with connections on a
dating app networks of cities and how
they're connected websites and how they
link to each other and so on with small
networks it can be easy to intuitively
understand what's going on like if I
said here's a group of coworkers and
connections represent mutual friendships
it wouldn't be hard to see like this is
the most popular person and this is the
least popular with only one friend if
you had to find how many mutual friends
these two people have no big deal you
can just count and see that's three but
when the networks get more complex we
need mathematical tools to help us
identify key things this could be like
which website should be ranked the
highest on the web which I mentioned
earlier it could be finding who is more
likely to spread a disease and a college
full of students for which people
involved in the 9/11 terrorist attacks
were most critical to the operation and
should be prioritized by law enforcement
yes they actually did this after 9/11
which I have discussed in a previous
video we need some mathematical
techniques in these cases so we can find
things that our eyes aren't always able
to when the connections get this chaotic
but let's do a matrices can reveal when
it comes to dating apps imagine this app
only has three men signed up that will
label 1 through 3 and 3 women labeled 4
through 6 and there
together as shown these are mutual
connections by the way like both people
swiping right and you'll know for now
there are no same-sex matches we see
this first guys matched with all three
women the second guy with two and the
third with one
now this graph provides a nice visual
for this situation but what we can also
do is make a table with six rows and six
columns for the six people and analyze
this instead we'll say if two people are
matched like person one in person for
our then in the square located and in
column 1 and row four we will put a 1
however since these are mutual
connections we need to also include a 1
in column 4 and Row 1 basically if one
matches with 4 then of course 4 has
matched with 1 so the data has to
reflect that so that means yes the
tables going to be symmetric about the
diagonal then if two people are not
connected like person 1 and 2 we'll put
a 0 in that square in this case column 1
in row 2 but of course we can't forget
column 2 and row 1 since no one matches
with themself the diagonal is going to
be all zeros and then these would be the
rest of the connections so if you want
to know whether person 5 and 2 are
connected by the table just go to column
5 and Row 2 or vice versa and see if
there's a one or a zero there from this
there are obvious things we can see like
for person 1 we can look down their
column or across the row and find they
have three matches in total because of
the three ones but we're going to use
some slightly more advanced math to
analyze this graph so instead of
considering this a table we're going to
call it a matrix but it's taking away
the gridlines but otherwise nothing has
changed when it comes to graphs this is
known as an adjacency matrix another way
to interpret this though is that it
tells us how many paths of length one
exists between any two nodes oh and for
the rest of this video when I say path I
just mean any sequence of edges that
joins a sequence of vertices basically
just a walk those no graph theory may
not like this because a path is usually
more specific but I am being generic
here okay so what's this really mean
we'll look at column 6 and row 1
we know this says that those two people
are matched no big deal
but it also means there is one path of
length one that exists between them
those are saying the same thing if we
put a dot at person 1 and can only
traverse one edge well there's one way
to get to person 6 that's what this one
represents on the other hand there are
zero ways to get from person 1 to person
2 in one edge if you start a person 1
there are paths to person 2 but they all
have a length of 2 which is not what we
were looking for but now what if I want
to see quickly how many mutual matches
two people have well if we look at
person 1 and 2 this isn't tough we see
there are 2 mutual matches however this
question of mutual matches is no
different than asking how many paths of
length 2 exists between person 1 and
person 2 well we just saw that the
answer is 2 as expected since it's the
same question again if I start at 1 and
go to 4 than 2 or 5 than 2 those two
paths mean two mutual connections the
cool thing though is that we can find
how many of these paths of length 2
exist between any two nodes by just
multiplying the adjacency matrix by
itself or squaring it we see for person
1 and 2 there are 2 mutual connections
so that checks out then if you look at
the graph for person 2 and 3 they have
no mutual connections and on the matrix
this checks out as well from column 3
and Row 2 having a 0 however person 1
and 3 are both connected to 6 and no one
else which we can also find in the
matrix and you get the idea but now what
would the diagonal mean well that's how
many paths of length 2 it exists between
a person and themselves
aka how many matches they have think
about it for person 1 if I start there
to get back to 1 in two edges I can go
into 4 than 1 5 than 1 or 6 and 1 3
options for the 3 connections this is
why we see a 3 there in the matrix
person 2 that has two matches and it
goes up
then if we multiply the new matrix by
the original the same as finding the
original cubed we get all the paths of
length 3 from one person to another see
the original matrix to some power tells
us how many paths of that length exist
between any two notes now if we have a
same-sex connection and link one to two
let's say all we have to do is add a 1
to the original matrix in the first
column second row and vice versa
see when implementing this in the
software we just have to make small
tweaks to the adjacency matrix and from
there squaring or cubing it tells us a
lot actually something I found
interesting was what the matrix cube
tells us specifically the diagonal for
one it tells us how many paths of length
three exists between a person and
themselves but looking at the graph a
length three path back to yourself tells
us there's a triangle there I'm not
going to explain this one in depth but
if you sum the numbers along the
diagonal also known as the trace of the
matrix and divide by six that tells you
how many triangles in total there are in
the network I just found that to be a
cool thing the matrix tells us which you
wouldn't think about it first then on
another topic something I haven't even
mentioned yet is machine learning and
neural networks which are coded with and
manipulated by matrix math
mathematically matrices are a huge
aspect of what allows the machine to
quote learn or in terms of security
there's an example of an older kind of
encryption method which is the hill
cipher this cipher incorporates matrix
operations in order to encrypt and
decrypt messages although no it's not a
modern encryption method and as much as
I'd love to keep going into depth on
different subjects this video is already
quite long so hopefully this show just
how powerful and impactful matrices are
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