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in a previous video I showed the basic
idea behind an eigenvector and
eigenvalue
but as a quick review when you have some
arbitrary two-by-two matrix and multiply
it by a vector let's say 1 comma 0 that
can be written as such you'll get out a
new vector or negative 1 comma negative
2 in this case instead of representing
vectors with arrows though I'll just be
using points which will represent the

00:30

end of the vector I'm using desmos so
it's just easier this way
but still we have an input and output
vector and you'll notice of the matrix
transformation cause the vector to scale
and also rotate that's usually what
happens put a random vector in here and
the output vector will likely have a
different length and angle in fact I'll
put a bunch of random input vectors on
the screen or really dots that represent
their ends and I'll apply the same
matrix multiplication to all of them at

00:58

once these are the new output vectors
and most of them are just scaled and
rotated versions of their associated
inputs but real quick if we rewind
you'll see that any point on these two
lines does not get rotated vectors on
these lines only get scaled and when a
matrix only scales vectors along some
line that is known as an eigen vector of
the matrix the scale factor or how much
larger the vector gets is known as the

01:32

eigenvalue you'll notice one of these
eigenvalues is 1 because when we apply
the transformation those vectors didn't
change in length if you want to rewind
it again you'll see the dots on the
other eigenvector all got three times as
far from the origin now if I change the
2 by 2 matrix to this aka a rotation
matrix then any input will be rotated 90
degrees so the question is what is the

02:03

eigenvector of this new matrix and by
the way 0 0 doesn't count
well considering an eigenvector is a
vector not rotated by the matrix
we can say that an eigenvector doesn't
exist here because every vector besides
zero zero gets rotated when you go
through the analysis though you do come
out with eigenvectors and eigenvalues
however their imaginary there isn't as
much visual intuition behind this what
we care about is this simple fact if the

02:33

eigenvalue slash vectors are imaginary
or just have an imaginary component then
any non-zero real input vector will be
rotated by some amount from the matrix
multiplication
it may be scaled it may not but some
rotation will definitely occur now the
first visual application we're going to
see the eigenvectors and eigenvalues is
the fibonacci sequence as most you know
this is the sequence that starts with 0
than 1 and you just add those to get the
next term then you add these to get the

03:03

next and this continues so
mathematically to get some term you add
the previous two now if I told you this
number as part of that sequence and
asked for the next number there's a nice
visual way to find it without just going
through the sequence itself the reason
for this visualization is because we can
express the equation through matrix
multiplication just by matrix rules
you'll see the first dot product gives
us FN minus 1 plus FN minus 2 equals FN
the same equation as above the second

03:37

dot product is trivial and just says FN
minus 1 plus 0 is itself we just need
this included so we can have a square
matrix so if I input any two neighboring
terms like 8 and 5 the resulting output
should just be the next term in the
sequence or 13 and the same FN minus 1
we input or 8 essentially the output is
just the rectangle slid over by 1 and
every matrix multiplication slides it
over by 1 again graphically this says if
I have the vector 8 comma 5 then the

04:08

matrix transformation should send it to
13 comma 8 which it does but let me do
that same transformation to several
Fibonacci coordinates where any points
or vectors represent two neighboring
terms
if I keep a copy of the inputs then we
can see the outputs just land on the
next vector in the sequence well we care
about most though is the fact that these
points all seem to be very close to this
line here because that is an eigenvector

04:39

of our matrix point really close to that
line pretty much just move further up
there's mostly no rotation the
associated eigenvalue is actually the
golden ratio or 1 plus root 5 over 2
that's how much further the points get
from the origin with each multiplication
roughly since the points here aren't
exactly on that line this does make it
easy to predict future terms though
because even way further into the
sequence the coordinates will pretty
much lie on this line it's the
eigenvector those points aren't getting

05:11

rotated off of it so if I take any point
and multiply its coordinates by the
golden ratio we get roughly the next
point up even for the early terms like
this the air is fairly small but as we
zoom out and let the term number go to
infinity the air goes to zero and as you
can see the points start to get
extremely close to that eigenvector this
means if I gave you some number way
further up the sequence and asked for
the next you just multiply by the golden
ratio and you will find it to nearly

05:43

exact precision what I also find really
interesting though is what happens when
we apply that same matrix transformation
to a bunch of random points think of
each of these as the first two numbers
in a new Fibonacci sequence with the
same arithmetic but a different
beginning if I apply the matrix
multiplication we get these points
representing the second pair of numbers
in the sequence then another
transformation gets us this set and as
we keep going the points all get closer

06:14

and closer to the eigenvector meaning
they all start to become scaled by the
golden ratio here I'll zoom out a bit so
you can see where they're at just after
four multiplications mathematically this
tells us whether we start at 0 and 1 or
3 and 17 or any other two positive
numbers so long as we use the same
formula and add the two previous
terms to get the next the numbers will
eventually just become a golden ratio
multiple of the previous that ratio is

06:45

built into the arithmetic and also this
matrix and its eigenvalues not the
starting terms themselves now for the
people who want a more real-world
application this next examples for you
because it combines calculus with
complex numbers with eigen stuff real
quick we first have to answer a few
questions though let's say I put a point
randomly on the XY plane and tell you it
will move as some function of time my
first question is what values of DX DT

07:15

and dy DT will cause it to move to the
right well if you've taken calculus this
is pretty easy DX DT has to be positive
while dy DT has to be zero that'll cause
a positive change in X but no change in
Y which is exactly what we want if I
want the particle to move up then dy DT
has to be positive while DX DT is zero
and you get the idea
but the slightly harder question is what
if I want to move towards or away from
the origin well in this case dy DT has

07:46

to be 1/2 of DX DT we can see this is we
think about those two velocities as a
coordinate because dy DT will be 1/2 of
DX DT when that coordinate is on this
line the same line that passes through
the particle itself and the origin the
equation for this is y equals 1/2 X just
to show the visuals if DX DT is 1 and dy
DT is 0.5 like this corner shows then
the vector sum yields this here which
points directly away from the origin and
that's the direction the particle would

08:16

move any of these values will give us
the same thing though while negative
velocities will result in motion towards
the origin ok now we're ready for the
applications let's say there's some
population of foxes we'll label
f and a population of rabbits with label
are these variables will represent the
number of each of those animals at any
given time then let's say that the
population of foxes changes at a rate
equal to its current population this
makes some sense in that the more foxes

08:48

you have
the faster they'll multiply but we are
assuming they live forever and there's
no upper bound on population because it
will simplify things if you've completed
calc one you know this leads to
exponential growth now the population of
rabbits will grow at a rate three times
their current population much faster
than the foxes however they will die or
be eaten at a rate equal to how many
foxes there are this makes things more
difficult but overall the goals to see
what will happen over time if we start

09:18

with some arbitrary number of both
animals now graphically just consider
the x axis to be rabbit population at
any time and the y axis will be foxes if
we say there are initially 10 rabbits
and 10 foxes that's represented as a
point at 10 comma 10 if we plug those
numbers into the equations then we get
out the derivatives which tells us the
rabbits will be increasing at twice the
rate as the Foxes just at that time this
is just like before so if we put the

09:48

dr/dt and the df/dt component at that
point basically like velocities the
vector sum will tell us that it will
move up and to the right just at that
moment this just means both the
populations are increasing because even
though the rabbits are being eaten that
are reproducing fast enough to increase
in population don't worry about the
length of the vector though just the
direction now had the rabbit population
start at 2 instead then plugging those
into the equations tells us that the
rabbits will decrease in number

10:19

essentially there's not enough of them
now and they'll die out faster than they
can reproduce so plotting those vector
components tells us that at that instant
the point will move up and to the left
it looks like we're headed towards 0
rabbits if I did this for every point we
make kind of like a slope field but it's
really called a phase plane for a system
of differential equations if I start at
let's say 10 comma 10 we just followed
the arrows and see how the population
will change over time starting at 2

10:49

comma 10 we see that it eventually does
reach 0 rabbits and these would just be
a bunch of other scenarios depending on
where you start
what you're seeing are the solutions
that set of differential equations for
different initial conditions but making
a phase plane one point a time is way
too tedious so let's look at this
another way I'm gonna take the
coefficients and put them into a matrix
that'll be multiplied by the two
variables R and F I'll call the
derivatives just R Prime and F Prime and
my matrix multiplication you'll see

11:21

these are the same thing now we can
apply matrix transformations from before
but this time notice that the inputs are
the rabbit and fox populations those
were the points you're seeing however
applying the matrix multiplication will
output velocities in a way or rates of
change of the population not new ones
here I'll put ten ten in a different
color
and do the multiplication while keeping
a copy of it at ten ten here we have the
outputs but again these represent the

11:52

instantaneous rates of change for their
Associated inputs as in at ten ten we'd
have an R prime of twenty and F prime of
ten if I proportionally shrink these and
put them on top of the input then this
tells us exactly what we saw before that
if you put ten rabbits and ten foxes in
our simulation and hit play at that
moment both populations would increase
but rabbits would be increasing twice as
fast now did you notice the eigenvector

12:23

though if I rewind you'll find anything
on this line or the x axis is not
rotated this means any inputs on those
lines will also have velocity values on
those lines and remember from before
when that was the case and that motion
would be along that line this means if
we have nine rabbits and zero foxes or
let's say three rabbits and six foxes as
long as we're on those eigenvectors and

12:53

let the simulation run the population
values will stay on those lines forever
see the bottom eigenvector is obvious if
you start with only rabbits then after
some time you'll have more rabbits no
surprise but the top one is kind of like
a sweet spot where you start with twice
as many fall
as rabbits and over time that ratio
never changes there's just enough foxes
where the rabbits don't take over in
population but not enough where the
rabbits die out completely everything
stays in proportion both of these have

13:26

positive eigenvalues which is why the
points move away from the origin and not
towards it but just these eigenvectors
are enough for us to approximate the
evolution of the system for any initial
conditions because the flow from any
other initial populations kind of
follows the eigenvectors which is why
the phase plane look the way that it did
plus the top eigenvector divided this
into interesting regions any populations
located above it will result in no
rabbits eventually all these arrows lead

13:58

to the y axis this is where the foxes
quote win they kill off the rabbits
faster than the rabbits can reproduce
but below that eigenvector and above the
other is where the rabbit sort of win
really it means over time they out
populate the Foxes more and more they're
still being eaten but they are
reproducing much faster and the
eigenvector is again the sweet spot
where everything stays in proportion
this is why before we saw some paths fly
off to the right and others turn towards

14:31

the left it was all dependent on which
side of the eigenvector we started on
and to add some dynamics here you'll see
if I increase this number and make it so
the rabbits reproduce faster then the
eigenvector moves up increasing the
rabbits wind section as expected if I
make the foxes reproduce faster then it
goes back down increasing that foxes
wind region the most well-known use of
this though is with masses on a spring

15:01

because instead of plotting animal
populations will plot position versus
velocity if I pull the mass to the left
but give it no initial velocity that
corresponds to this point on the graph
if I let go assuming no damping it will
oscillate back and forth forever on the
face plane this corresponds to circling
the origin between
axis placement no velocity to max
velocity no displacement and so on
forever now the equation that represents
this is MA or MV prime equals negative

15:33

KX minus BV where K is the spring
constant and B is the damping
coefficient currently the mass and
spring constant are 1 while B is 0 so
this is the actual equation and as you
guys probably know the derivative of
position is velocity now why am I
writing things like this because look we
have two derivatives some coefficients
and those same variables without the
prime just like before it's the same
type of problem which comes down to
finding eigen things however in this

16:04

case the eigen values are imaginary thus
so are the eigen vectors now do you guys
remember before when I said that
imaginary eigen values correspond to
rotation well here it is again in a much
more visual way for these systems of
differential equations purely imaginary
eigen values mean only rotation no decay
to equilibrium or diverging away from it
now if I increase the spring constant
and make the spring stronger we still
expect no decay but there should be some

16:35

higher velocities so what what happens
as I increment it now we get more of an
oval shape showing that the max velocity
has increased but the max position does
not since it starts from the same point
the eigen value is still imaginary
though meaning only rotation and no
decay if we have some light damping like
put the block in some fluid we expect
the block to oscillate but decayed
equilibrium in the process so first

17:07

watch what happens to the phase plane as
I change that damping coefficient
you'll see it now spirals into the
origin aka the equilibrium where there's
no velocity or displacement which
matches what we expect with the decay
that comes from the damping from the
eigenvalue perspective we gain even more
intuition because on top of imaginary
components corresponding to rotation we
can now see that negative real
components correspond to decay as in the
system is stable and will go to

17:42

equilibrium eventually here I'll
increase the damping coefficient even
more where we see faster decay on the
phase plane and as expected a more
negative real component of our
eigenvalue and if it were possible to
have a positive damping coefficient here
which I'll change it to then that gives
us a positive real component which
corresponds to growth or instability and
this really means that the phase plane
will spiral away from equilibrium now so

18:15

far we've seen how eigenvalues and
eigenvectors can give us a real visual
sense for what can happen to a system or
a sequence after a long time or many
iterations this isn't always the case
but in many situations heigen vectors
and eigen values reveal that long term
behavior that's one reason why they're
so useful in the applications of
matrices video I showed an example of
this with a zombie apocalypse scenario
where in a quarantine area every hour
some percentage of humans turn into
zombies but there is a cure that

18:45

converts a percentage of zombies back to
humans since the new population of
humans and zombies after each hour can
be calculated with linear equations we
can change it into a matrix problem to
see what happens in the long run if I
use desmos instead we can see with
several starting points representing
initial populations of humans and
zombies the eigenvector reveals the
equilibrium of the system
and since the eigenvalue is 1 once the
populations are on that then they won't
change but everything else will move

19:17

towards it with each hour
so after many iterations the system will
converge to a point where there are
twice as many zombies as humans and
again it's the eigenvector that reveals
that long term behavior or let's say we
had several nodes in one way path
connecting them if you were to just
randomly walk this network all day and
want to know how much time you'd spend
at each node on average it becomes a
matrix problem because since everything
is random we can write out the
probabilities like if you're at a

19:48

there's a 100 percent chance of going to
be but once you're there you have a 50%
chance of going to C or D and you just
do this for every edge if we put those
probabilities into a matrix where you
can see like the chance of going from A
to B is a hundred percent but B to C is
50% it turns out that eigen vector of
that matrix reveals the percentage of
time you'll spend at each node if we
instead think of those nodes as web
sites and the edges as links between
them then you get the basic idea of how
Google ranks web pages yes in the

20:19

beginning the basic idea of Google
Pagerank was determining which site
you'd spend more time on if you randomly
clicked links all day and that ranking
matrix is really just an eigenvector but
it's not always about long term behavior
for example also from that previous
video I didn't mention something about
the dating app Network example this was
when three men labeled one through three
and three women four through six were
matched on a dating site as shown but
there were no same sex matches which
makes us a bipartite graph bipartite

20:49

just means the network can be separated
into two different sets such that each
one of these has no internal connections
now when we made the adjacency matrix
for this we're if like person 1 and 4
connected then we see a 1 in column 1
and row 4 as well as Row one in column 4
and if two people were not connected
there B is 0 giving us this what I
didn't say is that these were the
eigenvalues it turns out if a graph is
bipartite then the eigenvalues of this
matrix will come in plus and minus pairs

21:22

if we allow the same-sex match then this
is no longer bipartite and the
eigenvalues don't come in those pairs so
if I gave you a network like this it's
not obvious whether it's bipartite just
by looking at it but if I label the
nodes and make an adjacency matrix based
on those connections I find the
eigenvalues all come in plus and minus
pairs which tells us yes the graph is
bipartite and if we rearrange that
becomes more obvious looking at how
clustered a graph is also involves

21:53

eigenvalue analysis by the way and this
can reveal hidden patterns among very
chaotic graphs you can look up spectral
clustering to learn more about this
though
then eigenvalues show up in frequency
analysis as you're used to find the
natural frequency of a system of course
they appear in circuits do the
similarities between those and masses on
the spring eigenvalues can help us
understand how quickly a disease will
spread throughout a population they're
used in data compression and facial
recognition tons of physics applications
and of course way more than I can put in

22:23

one video but hopefully this gave you an
idea of the power and intuition behind
these things that I think for most
people are very dry when we learn them
in school and if you've made it this far
and want to expand your knowledge
further into a variety of topics I
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