WEBVTT
Kind: captions
Language: en
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let me pull out an old differential
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equations textbook that i learned from
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in college
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and let's turn to this funny little
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exercise in here that asks the reader to
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compute
00:00:08.480 --> 00:00:12.080
e to the power a t where a we're told is
00:00:12.080 --> 00:00:13.440
going to be a matrix
00:00:13.440 --> 00:00:15.200
and the insinuation seems to be that the
00:00:15.200 --> 00:00:18.240
result will also be a matrix
00:00:18.240 --> 00:00:20.000
it then offers several examples for what
00:00:20.000 --> 00:00:22.000
you might plug in for a
00:00:22.000 --> 00:00:23.680
now taken out of context putting a
00:00:23.680 --> 00:00:25.359
matrix into an exponent like this
00:00:25.359 --> 00:00:27.519
probably seems like total nonsense but
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what it refers to is an extremely
00:00:29.439 --> 00:00:30.720
beautiful operation
00:00:30.720 --> 00:00:32.320
and the reason it shows up in this book
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is that it's useful it's used to solve a
00:00:34.719 --> 00:00:36.480
very important class of differential
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equations
00:00:37.600 --> 00:00:39.760
in turn given that the universe is often
00:00:39.760 --> 00:00:41.440
written in the language of differential
00:00:41.440 --> 00:00:42.320
equations
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you see this pop up in physics all the
00:00:43.760 --> 00:00:46.399
time too especially in quantum mechanics
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where matrix exponents are littered
00:00:48.000 --> 00:00:49.280
throughout the place they play a
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particularly prominent role
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this has a lot to do with schrodinger's
00:00:52.559 --> 00:00:53.920
equation which we'll touch on a bit
00:00:53.920 --> 00:00:54.640
later
00:00:54.640 --> 00:00:56.160
and it may also help in understanding
00:00:56.160 --> 00:00:58.079
your romantic relationships but
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again all in due time
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a big part of the reason i want to cover
00:01:07.119 --> 00:01:09.040
this topic is that there is an extremely
00:01:09.040 --> 00:01:10.320
nice way to visualize
00:01:10.320 --> 00:01:12.400
what matrix exponents are actually doing
00:01:12.400 --> 00:01:14.400
using flow that not a lot of people seem
00:01:14.400 --> 00:01:15.439
to talk about
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but for the bulk of this chapter let's
00:01:17.040 --> 00:01:18.799
start by laying out what exactly the
00:01:18.799 --> 00:01:19.600
operation
00:01:19.600 --> 00:01:21.680
is and see if we can get a feel for what
00:01:21.680 --> 00:01:24.240
kinds of problems it helps us to solve
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the first thing you should know is that
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this is not some bizarre way to multiply
00:01:27.600 --> 00:01:28.560
the constant e
00:01:28.560 --> 00:01:31.040
by itself multiple times you would be
00:01:31.040 --> 00:01:32.799
right to call that nonsense
00:01:32.799 --> 00:01:34.640
the actual definition is related to a
00:01:34.640 --> 00:01:36.159
certain infinite polynomial for
00:01:36.159 --> 00:01:38.240
describing real number powers of e
00:01:38.240 --> 00:01:40.720
what we call its taylor series for
00:01:40.720 --> 00:01:42.479
example if i took the number 2 and
00:01:42.479 --> 00:01:44.240
plugged it into this polynomial
00:01:44.240 --> 00:01:47.040
then as you add more and more terms each
00:01:47.040 --> 00:01:48.880
of which looks like some power of 2
00:01:48.880 --> 00:01:54.240
divided by some factorial
00:01:54.240 --> 00:01:58.399
the sum approaches a number near 7.389
00:01:58.399 --> 00:02:01.920
and this number is precisely e times e
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if you increment this input by one then
00:02:04.399 --> 00:02:06.000
somewhat miraculously no matter where
00:02:06.000 --> 00:02:07.040
you started from
00:02:07.040 --> 00:02:09.119
the effect on the output is always to
00:02:09.119 --> 00:02:12.160
multiply it by another factor of e
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for reasons that you're going to see in
00:02:13.520 --> 00:02:15.440
a bit mathematicians became interested
00:02:15.440 --> 00:02:17.040
in plugging all kinds of things into
00:02:17.040 --> 00:02:18.080
this polynomial
00:02:18.080 --> 00:02:20.000
things like complex numbers and for our
00:02:20.000 --> 00:02:22.080
purposes today matrices
00:02:22.080 --> 00:02:23.920
even when those objects do not
00:02:23.920 --> 00:02:26.400
immediately make sense as exponents
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what some authors do is give this
00:02:28.080 --> 00:02:29.680
infinite polynomial the name
00:02:29.680 --> 00:02:32.319
x when you plug in more exotic inputs
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it's a gentle nod to the connection that
00:02:34.400 --> 00:02:36.319
this has to exponential functions in the
00:02:36.319 --> 00:02:37.440
case of real numbers
00:02:37.440 --> 00:02:39.040
even though obviously these inputs don't
00:02:39.040 --> 00:02:40.720
make sense as exponents
00:02:40.720 --> 00:02:43.519
however an equally common convention is
00:02:43.519 --> 00:02:45.200
to give a much less gentle nod to the
00:02:45.200 --> 00:02:46.959
connection and just abbreviate the whole
00:02:46.959 --> 00:02:48.879
thing as e to the power of whatever
00:02:48.879 --> 00:02:50.239
object you're plugging in
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whether that's a complex number or a
00:02:51.920 --> 00:02:53.920
matrix or all sorts of more exotic
00:02:53.920 --> 00:02:54.959
objects
00:02:54.959 --> 00:02:56.959
so while this equation is a theorem for
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real numbers it's a definition for more
00:02:59.280 --> 00:03:00.800
exotic inputs
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cynically you could call this a blatant
00:03:02.800 --> 00:03:04.560
abuse of notation
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more charitably you might view it as an
00:03:06.239 --> 00:03:07.920
example of the beautiful cycle between
00:03:07.920 --> 00:03:10.560
discovery and invention in math
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in either case plugging in a matrix even
00:03:12.480 --> 00:03:14.080
to a polynomial might seem a little
00:03:14.080 --> 00:03:14.640
strange
00:03:14.640 --> 00:03:16.720
so let's be clear on what we mean here
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the matrix has to have the same number
00:03:18.560 --> 00:03:20.319
of rows and columns
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that way you can multiply it by itself
00:03:22.159 --> 00:03:23.920
according to the usual rules of matrix
00:03:23.920 --> 00:03:25.200
multiplication
00:03:25.200 --> 00:03:27.920
this is what we mean by squaring it
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similarly if you were to take
00:03:29.360 --> 00:03:31.440
that result and then multiply it by the
00:03:31.440 --> 00:03:33.280
original matrix again
00:03:33.280 --> 00:03:35.120
this is what we mean by cubing the
00:03:35.120 --> 00:03:35.880
matrix
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[Music]
00:03:37.760 --> 00:03:39.680
if you carry on like this you can take
00:03:39.680 --> 00:03:41.840
any whole number power of a matrix it's
00:03:41.840 --> 00:03:43.200
perfectly sensible
00:03:43.200 --> 00:03:44.879
in this context powers still mean
00:03:44.879 --> 00:03:46.239
exactly what you would expect
00:03:46.239 --> 00:03:54.000
repeated multiplication
00:03:54.000 --> 00:03:56.000
each term in this polynomial is scaled
00:03:56.000 --> 00:03:58.239
by 1 divided by some factorial
00:03:58.239 --> 00:03:59.840
and with matrices all that means is that
00:03:59.840 --> 00:04:01.840
you multiply each component by that
00:04:01.840 --> 00:04:03.040
number
00:04:03.040 --> 00:04:04.799
likewise it always makes sense to add
00:04:04.799 --> 00:04:06.640
together two matrices this is something
00:04:06.640 --> 00:04:07.599
that you again do
00:04:07.599 --> 00:04:10.560
term by term the astute among you might
00:04:10.560 --> 00:04:12.319
ask how sensible it is to take this out
00:04:12.319 --> 00:04:13.280
to infinity
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which would be a great question one that
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i'm largely going to postpone the answer
00:04:16.639 --> 00:04:17.199
to
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but i can show you one pretty fun
00:04:18.639 --> 00:04:20.320
example here now
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take this 2x2 matrix that has negative
00:04:22.720 --> 00:04:23.520
pi and pi
00:04:23.520 --> 00:04:25.600
sitting off its diagonal entries let's
00:04:25.600 --> 00:04:27.120
see what the sum gives
00:04:27.120 --> 00:04:29.680
the first term is the identity matrix
00:04:29.680 --> 00:04:30.960
this is actually what we mean by
00:04:30.960 --> 00:04:32.720
definition when we raise a matrix to the
00:04:32.720 --> 00:04:34.240
zeroth power
00:04:34.240 --> 00:04:36.160
then we add the matrix itself which
00:04:36.160 --> 00:04:38.720
gives us the pi off the diagonal terms
00:04:38.720 --> 00:04:41.199
and then add half of the matrix squared
00:04:41.199 --> 00:04:43.360
and continuing on i'll have the computer
00:04:43.360 --> 00:04:43.680
keep
00:04:43.680 --> 00:04:45.440
adding more and more terms each of which
00:04:45.440 --> 00:04:47.440
requires taking one more matrix product
00:04:47.440 --> 00:04:48.639
to get the new power
00:04:48.639 --> 00:04:50.960
and then adding it to a running tally
00:04:50.960 --> 00:04:52.240
and as it keeps going
00:04:52.240 --> 00:04:54.400
it seems to be approaching a stable
00:04:54.400 --> 00:04:56.720
value which is around negative 1
00:04:56.720 --> 00:04:59.280
times the identity matrix in this sense
00:04:59.280 --> 00:05:00.479
we say the infinite sum
00:05:00.479 --> 00:05:03.199
equals that negative identity by the end
00:05:03.199 --> 00:05:04.800
of this video my hope is that this
00:05:04.800 --> 00:05:05.840
particular fact
00:05:05.840 --> 00:05:08.080
comes to make total sense to you for any
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of you familiar with euler's famous
00:05:09.600 --> 00:05:10.320
identity
00:05:10.320 --> 00:05:11.919
this is essentially the matrix version
00:05:11.919 --> 00:05:14.000
of that it turns out that
00:05:14.000 --> 00:05:15.919
in general no matter what matrix you
00:05:15.919 --> 00:05:17.759
start with as you add more and more
00:05:17.759 --> 00:05:18.320
terms
00:05:18.320 --> 00:05:20.160
you eventually approach some stable
00:05:20.160 --> 00:05:21.440
value though
00:05:21.440 --> 00:05:22.800
sometimes it can take quite a while
00:05:22.800 --> 00:05:26.479
before you get there
00:05:26.479 --> 00:05:28.560
just seeing the definition like this in
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isolation
00:05:29.600 --> 00:05:31.919
raises all kinds of questions most
00:05:31.919 --> 00:05:32.960
notably
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why would mathematicians and physicists
00:05:34.880 --> 00:05:36.479
be interested in torturing their poor
00:05:36.479 --> 00:05:37.680
matrices this way
00:05:37.680 --> 00:05:40.080
what problems are they trying to solve
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and if you're anything like me
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a new operation is only satisfying when
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you have a clear view
00:05:44.479 --> 00:05:46.960
of what it's trying to do some sense of
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how to predict the output based on the
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input before you actually crunch the
00:05:50.320 --> 00:05:51.199
numbers
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how on earth could you have predicted
00:05:52.880 --> 00:05:54.320
that the matrix with pi off the
00:05:54.320 --> 00:05:55.360
diagonals
00:05:55.360 --> 00:05:57.199
results in a negative identity matrix
00:05:57.199 --> 00:05:58.880
like this
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often in math you should view the
00:06:00.319 --> 00:06:02.479
definition not as a starting point but
00:06:02.479 --> 00:06:03.680
as a target
00:06:03.680 --> 00:06:05.600
contrary to the structure of textbooks
00:06:05.600 --> 00:06:07.360
mathematicians do not start by making
00:06:07.360 --> 00:06:08.240
definitions
00:06:08.240 --> 00:06:09.759
and then listing a lot of theorems and
00:06:09.759 --> 00:06:11.120
proving them and then showing some
00:06:11.120 --> 00:06:12.080
examples
00:06:12.080 --> 00:06:13.600
the process of discovering math
00:06:13.600 --> 00:06:15.360
typically goes the other way around
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they start by chewing on specific
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problems and then generalizing those
00:06:19.280 --> 00:06:20.160
problems
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then coming up with constructs that
00:06:21.680 --> 00:06:23.840
might be helpful in those general cases
00:06:23.840 --> 00:06:25.759
and only then do you write down a new
00:06:25.759 --> 00:06:27.039
definition or
00:06:27.039 --> 00:06:30.160
extend an old one as to what sorts of
00:06:30.160 --> 00:06:32.000
specific examples might motivate matrix
00:06:32.000 --> 00:06:32.880
exponents
00:06:32.880 --> 00:06:35.199
two come to mind one involving
00:06:35.199 --> 00:06:36.880
relationships and the other quantum
00:06:36.880 --> 00:06:38.000
mechanics
00:06:38.000 --> 00:06:43.039
let's start with relationships
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suppose we have two lovers let's call
00:06:44.880 --> 00:06:46.319
them romeo and juliet
00:06:46.319 --> 00:06:48.639
and let's let x represent juliet's love
00:06:48.639 --> 00:06:50.400
for romeo
00:06:50.400 --> 00:06:53.599
and y represent his love for her both of
00:06:53.599 --> 00:06:55.199
which are going to be values that change
00:06:55.199 --> 00:06:56.720
with time
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this is an example that we actually
00:06:58.080 --> 00:06:59.919
touched on in chapter one it's based on
00:06:59.919 --> 00:07:01.360
a steven strogatz article
00:07:01.360 --> 00:07:03.599
but it's okay if you didn't see that the
00:07:03.599 --> 00:07:05.520
way their relationship works is that the
00:07:05.520 --> 00:07:07.599
rate at which juliet's love for romeo
00:07:07.599 --> 00:07:08.400
changes
00:07:08.400 --> 00:07:11.039
the derivative of this value is equal to
00:07:11.039 --> 00:07:12.000
negative one
00:07:12.000 --> 00:07:14.960
times romeo's love for her so in other
00:07:14.960 --> 00:07:15.440
words
00:07:15.440 --> 00:07:17.199
when romeo is expressing cool
00:07:17.199 --> 00:07:19.039
disinterest that's when juliet's
00:07:19.039 --> 00:07:20.639
feelings actually increase
00:07:20.639 --> 00:07:23.360
whereas if he becomes too infatuated her
00:07:23.360 --> 00:07:26.880
interest will start to fade
00:07:26.880 --> 00:07:28.800
romeo on the other hand is the opposite
00:07:28.800 --> 00:07:30.000
the rate of change of his
00:07:30.000 --> 00:07:32.319
love is equal to the size of juliet's
00:07:32.319 --> 00:07:33.039
love
00:07:33.039 --> 00:07:34.960
so while juliet is mad at him his
00:07:34.960 --> 00:07:38.880
affections tend to decrease
00:07:38.880 --> 00:07:40.800
whereas if she loves him that's when his
00:07:40.800 --> 00:07:42.479
feelings grow
00:07:42.479 --> 00:07:44.319
of course neither one of these numbers
00:07:44.319 --> 00:07:45.520
is holding still
00:07:45.520 --> 00:07:48.000
as romeo's love increases in response to
00:07:48.000 --> 00:07:48.639
juliet
00:07:48.639 --> 00:07:51.039
her equation continues to apply and
00:07:51.039 --> 00:07:53.199
drives her love down
00:07:53.199 --> 00:07:55.360
both of these equations always apply
00:07:55.360 --> 00:07:57.440
from each infinitesimal point in time to
00:07:57.440 --> 00:07:58.160
the next
00:07:58.160 --> 00:08:00.479
so every slight change to one value
00:08:00.479 --> 00:08:02.160
immediately influences the rate of
00:08:02.160 --> 00:08:04.160
change of the other
00:08:04.160 --> 00:08:05.919
this is a system of differential
00:08:05.919 --> 00:08:07.759
equations it's a puzzle
00:08:07.759 --> 00:08:09.919
where your challenge is to find explicit
00:08:09.919 --> 00:08:11.280
functions for x of t
00:08:11.280 --> 00:08:13.440
and y of t that make both of these
00:08:13.440 --> 00:08:15.440
expressions true
00:08:15.440 --> 00:08:17.440
now as systems of differential equations
00:08:17.440 --> 00:08:19.919
go this one is on the simpler side
00:08:19.919 --> 00:08:21.840
enough so that many calculus students
00:08:21.840 --> 00:08:24.240
could probably just guess at an answer
00:08:24.240 --> 00:08:26.400
but keep in mind it's not enough to find
00:08:26.400 --> 00:08:28.080
some pair of functions that makes this
00:08:28.080 --> 00:08:28.960
true
00:08:28.960 --> 00:08:30.720
if you want to actually predict where
00:08:30.720 --> 00:08:32.640
romeo and juliet end up after some
00:08:32.640 --> 00:08:33.599
starting point
00:08:33.599 --> 00:08:34.640
you have to make sure that your
00:08:34.640 --> 00:08:36.479
functions match the initial set of
00:08:36.479 --> 00:08:37.360
conditions
00:08:37.360 --> 00:08:40.479
at time t equals 0. more to the point
00:08:40.479 --> 00:08:42.080
our actual goal today is to
00:08:42.080 --> 00:08:43.919
systematically solve more general
00:08:43.919 --> 00:08:45.360
versions of this equation
00:08:45.360 --> 00:08:47.200
without guessing and checking and it's
00:08:47.200 --> 00:08:48.800
that question that leads us to matrix
00:08:48.800 --> 00:08:50.560
exponents
00:08:50.560 --> 00:08:52.160
very often when you have multiple
00:08:52.160 --> 00:08:53.600
changing values like this
00:08:53.600 --> 00:08:55.440
it's helpful to package them together as
00:08:55.440 --> 00:08:56.880
coordinates of a single point in a
00:08:56.880 --> 00:08:58.560
higher dimensional space
00:08:58.560 --> 00:09:00.560
so for romeo and juliet think of their
00:09:00.560 --> 00:09:02.959
relationship as a point in a 2d space
00:09:02.959 --> 00:09:06.839
the x-coordinate capturing juliet's
00:09:06.839 --> 00:09:08.240
feelings
00:09:08.240 --> 00:09:13.120
and the y coordinate capturing romeos
00:09:13.120 --> 00:09:14.720
sometimes it's helpful to picture this
00:09:14.720 --> 00:09:16.800
state as an arrow from the origin
00:09:16.800 --> 00:09:18.800
other times just as a point all that
00:09:18.800 --> 00:09:20.720
really matters is that it encodes two
00:09:20.720 --> 00:09:21.600
numbers
00:09:21.600 --> 00:09:23.519
and moving forward we'll be writing that
00:09:23.519 --> 00:09:25.200
as a column vector
00:09:25.200 --> 00:09:27.040
and of course this is all a function of
00:09:27.040 --> 00:09:28.320
time
00:09:28.320 --> 00:09:30.080
you might picture the rate of change of
00:09:30.080 --> 00:09:31.760
this state the thing that packages
00:09:31.760 --> 00:09:33.200
together the derivative of x
00:09:33.200 --> 00:09:35.360
and the derivative of y as a kind of
00:09:35.360 --> 00:09:36.560
velocity vector
00:09:36.560 --> 00:09:38.720
in this state space something that tugs
00:09:38.720 --> 00:09:40.240
at our point in some direction
00:09:40.240 --> 00:09:42.080
and with some magnitude that indicates
00:09:42.080 --> 00:09:45.440
how quickly it's changing
00:09:45.440 --> 00:09:47.760
remember the rule here is that the rate
00:09:47.760 --> 00:09:48.720
of change of x
00:09:48.720 --> 00:09:51.040
is negative y and the rate of change of
00:09:51.040 --> 00:09:52.000
y is
00:09:52.000 --> 00:09:55.040
x set up as vectors like this we could
00:09:55.040 --> 00:09:56.560
rewrite the right hand side of this
00:09:56.560 --> 00:09:57.279
equation
00:09:57.279 --> 00:09:59.839
as a product of this matrix with the
00:09:59.839 --> 00:10:00.800
original vector
00:10:00.800 --> 00:10:04.640
x y the top row encodes juliet's rule
00:10:04.640 --> 00:10:07.680
and the bottom row encodes romeo's rule
00:10:07.680 --> 00:10:09.680
so what we have here is a differential
00:10:09.680 --> 00:10:11.360
equation telling us that the rate of
00:10:11.360 --> 00:10:12.959
change of some vector
00:10:12.959 --> 00:10:16.839
is equal to a certain matrix times
00:10:16.839 --> 00:10:18.800
itself
00:10:18.800 --> 00:10:20.560
in a moment we'll talk about how matrix
00:10:20.560 --> 00:10:22.399
exponentiation solves this kind of
00:10:22.399 --> 00:10:23.279
equation
00:10:23.279 --> 00:10:24.640
but before that let me show you a
00:10:24.640 --> 00:10:26.079
simpler way that we can solve this
00:10:26.079 --> 00:10:27.360
particular system
00:10:27.360 --> 00:10:29.680
one that uses pure geometry and it helps
00:10:29.680 --> 00:10:31.440
set the stage for visualizing matrix
00:10:31.440 --> 00:10:32.000
exponents
00:10:32.000 --> 00:10:35.360
a bit later this matrix from our system
00:10:35.360 --> 00:10:38.800
is a 90 degree rotation matrix
00:10:38.800 --> 00:10:40.320
for any of you rusty on how to think
00:10:40.320 --> 00:10:42.480
about matrices as transformations
00:10:42.480 --> 00:10:44.000
there's a video all about it on this
00:10:44.000 --> 00:10:46.160
channel a series really
00:10:46.160 --> 00:10:48.079
the basic idea is that when you multiply
00:10:48.079 --> 00:10:49.680
a matrix by the vector 1
00:10:49.680 --> 00:10:53.839
0 it pulls out the first column
00:10:53.839 --> 00:10:56.880
and similarly if you multiply it by 0 1
00:10:56.880 --> 00:10:59.920
that pulls out the second column what
00:10:59.920 --> 00:11:01.600
this means is that when you look at a
00:11:01.600 --> 00:11:02.399
matrix
00:11:02.399 --> 00:11:04.720
you can read its columns as telling you
00:11:04.720 --> 00:11:06.959
what it does to these two vectors
00:11:06.959 --> 00:11:09.680
known as the basis vectors the way it
00:11:09.680 --> 00:11:11.200
acts on any other vector
00:11:11.200 --> 00:11:13.360
is a result of scaling and adding these
00:11:13.360 --> 00:11:14.880
two basis results
00:11:14.880 --> 00:11:17.920
by that vector's coordinates so looking
00:11:17.920 --> 00:11:19.440
back at the matrix from our system
00:11:19.440 --> 00:11:21.200
notice how from its columns we can tell
00:11:21.200 --> 00:11:23.680
it takes the first basis vector to 0 1
00:11:23.680 --> 00:11:26.959
and the second to negative 1 0 hence why
00:11:26.959 --> 00:11:28.560
i'm calling it the 90 degree rotation
00:11:28.560 --> 00:11:31.040
matrix
00:11:31.040 --> 00:11:32.959
what it means for our equation is that
00:11:32.959 --> 00:11:35.040
it's saying wherever romeo and juliet
00:11:35.040 --> 00:11:35.440
are
00:11:35.440 --> 00:11:38.000
in this state space their rate of change
00:11:38.000 --> 00:11:40.480
has to look like a 90 degree rotation
00:11:40.480 --> 00:11:43.040
of this position vector the only way
00:11:43.040 --> 00:11:44.640
velocity can permanently be
00:11:44.640 --> 00:11:46.640
perpendicular to position like this
00:11:46.640 --> 00:11:48.560
is when you rotate around the origin in
00:11:48.560 --> 00:11:49.760
circular motion
00:11:49.760 --> 00:11:51.200
never growing or shrinking because the
00:11:51.200 --> 00:11:52.880
rate of change has no component
00:11:52.880 --> 00:11:57.040
in the direction of the position
00:11:57.040 --> 00:11:59.360
more specifically since the length of
00:11:59.360 --> 00:12:00.880
this velocity vector
00:12:00.880 --> 00:12:03.440
equals the length of the position vector
00:12:03.440 --> 00:12:05.360
then for each unit of time
00:12:05.360 --> 00:12:07.600
the distance that this covers is equal
00:12:07.600 --> 00:12:09.600
to one radius's worth of arc length
00:12:09.600 --> 00:12:10.000
along
00:12:10.000 --> 00:12:13.760
that circle in other words it rotates at
00:12:13.760 --> 00:12:15.920
1 radian per unit time
00:12:15.920 --> 00:12:18.320
so in particular it would take 2 pi
00:12:18.320 --> 00:12:19.279
units of time
00:12:19.279 --> 00:12:22.800
to make a full revolution
00:12:22.800 --> 00:12:24.079
if you want to describe this kind of
00:12:24.079 --> 00:12:26.639
rotation with a formula we can use a
00:12:26.639 --> 00:12:28.320
more general rotation matrix
00:12:28.320 --> 00:12:31.120
which looks like this again we can read
00:12:31.120 --> 00:12:32.560
it in terms of the columns
00:12:32.560 --> 00:12:34.079
notice how the first column tells us
00:12:34.079 --> 00:12:36.079
that it takes that first basis vector
00:12:36.079 --> 00:12:39.839
to cosine of t sine of t
00:12:39.839 --> 00:12:41.440
and the second column tells us that it
00:12:41.440 --> 00:12:43.040
takes the second basis vector
00:12:43.040 --> 00:12:45.760
to negative sine of t cosine of t both
00:12:45.760 --> 00:12:46.160
of which
00:12:46.160 --> 00:12:48.000
are consistent with rotating by t
00:12:48.000 --> 00:12:49.600
radians
00:12:49.600 --> 00:12:51.519
so to solve the system if you want to
00:12:51.519 --> 00:12:53.600
predict where romeo and juliet end up
00:12:53.600 --> 00:12:56.560
after t units of time you can multiply
00:12:56.560 --> 00:12:57.600
this matrix
00:12:57.600 --> 00:13:00.880
by their initial state the active
00:13:00.880 --> 00:13:02.240
viewers among you might also enjoy
00:13:02.240 --> 00:13:03.920
taking a moment to pause and confirm
00:13:03.920 --> 00:13:05.760
that the explicit formulas you get out
00:13:05.760 --> 00:13:06.880
of this for x of t
00:13:06.880 --> 00:13:09.279
and y of t really do satisfy the system
00:13:09.279 --> 00:13:10.639
of differential equations that we
00:13:10.639 --> 00:13:16.370
started with
00:13:16.370 --> 00:13:17.839
[Music]
00:13:17.839 --> 00:13:19.519
the mathematician in you might wonder if
00:13:19.519 --> 00:13:21.440
it's possible to solve not just this
00:13:21.440 --> 00:13:22.560
specific system
00:13:22.560 --> 00:13:24.240
but equations like it for any other
00:13:24.240 --> 00:13:26.959
matrix no matter what its coefficients
00:13:26.959 --> 00:13:29.200
to ask this question is to set yourself
00:13:29.200 --> 00:13:31.519
up to rediscover matrix exponents
00:13:31.519 --> 00:13:33.200
the main goal for today is for you to
00:13:33.200 --> 00:13:34.959
understand how this equation
00:13:34.959 --> 00:13:36.720
lets you intuitively picture the
00:13:36.720 --> 00:13:38.480
operation which we write as e
00:13:38.480 --> 00:13:40.880
raised to a matrix and on the flip side
00:13:40.880 --> 00:13:42.560
how being able to compute matrix
00:13:42.560 --> 00:13:43.360
exponents
00:13:43.360 --> 00:13:46.320
lets you explicitly solve this equation
00:13:46.320 --> 00:13:48.079
a much less whimsical example is
00:13:48.079 --> 00:13:49.760
schrodinger's famous equation
00:13:49.760 --> 00:13:51.279
which is the fundamental equation
00:13:51.279 --> 00:13:53.120
describing how systems in quantum
00:13:53.120 --> 00:13:54.240
mechanics change
00:13:54.240 --> 00:13:57.040
over time it looks pretty intimidating
00:13:57.040 --> 00:13:58.639
and i mean it's quantum mechanics so of
00:13:58.639 --> 00:13:59.519
course it will
00:13:59.519 --> 00:14:00.800
but it's actually not that different
00:14:00.800 --> 00:14:02.880
from the romeo julia setup
00:14:02.880 --> 00:14:04.800
this symbol here refers to a certain
00:14:04.800 --> 00:14:06.639
vector it's a vector that packages
00:14:06.639 --> 00:14:08.160
together all the information you might
00:14:08.160 --> 00:14:09.440
care about in a system
00:14:09.440 --> 00:14:11.279
like the various particles positions and
00:14:11.279 --> 00:14:13.680
momenta it's analogous to our simpler 2d
00:14:13.680 --> 00:14:15.519
vector that encoded all the information
00:14:15.519 --> 00:14:17.680
about romeo and juliet
00:14:17.680 --> 00:14:19.600
the equation says that the rate at which
00:14:19.600 --> 00:14:21.279
this state vector changes
00:14:21.279 --> 00:14:24.399
looks like a certain matrix times itself
00:14:24.399 --> 00:14:25.839
there are a number of things that make
00:14:25.839 --> 00:14:27.440
schrodinger's equation notably more
00:14:27.440 --> 00:14:28.320
complicated
00:14:28.320 --> 00:14:29.839
but in the back of your mind you might
00:14:29.839 --> 00:14:31.440
think of it as a target point that you
00:14:31.440 --> 00:14:32.800
and i can build up to
00:14:32.800 --> 00:14:34.800
with simpler examples like romeo and
00:14:34.800 --> 00:14:36.880
juliet offering more friendly stepping
00:14:36.880 --> 00:14:39.360
stones along the way
00:14:39.360 --> 00:14:41.279
actually the simplest example which is
00:14:41.279 --> 00:14:43.760
tied to ordinary real number powers of e
00:14:43.760 --> 00:14:45.680
is the one dimensional case this is when
00:14:45.680 --> 00:14:47.600
you have a single changing value
00:14:47.600 --> 00:14:49.199
and its rate of change equals some
00:14:49.199 --> 00:14:50.959
constant times itself
00:14:50.959 --> 00:14:52.880
so the bigger the value the faster it
00:14:52.880 --> 00:14:54.959
grows
00:14:54.959 --> 00:14:56.160
most people are more comfortable
00:14:56.160 --> 00:14:58.480
visualizing this with a graph where the
00:14:58.480 --> 00:14:59.920
higher the value of the graph the
00:14:59.920 --> 00:15:01.120
steeper its slope
00:15:01.120 --> 00:15:03.120
resulting in this ever steepening upward
00:15:03.120 --> 00:15:05.120
curve just keep in mind that when we get
00:15:05.120 --> 00:15:06.639
to higher dimensional variants
00:15:06.639 --> 00:15:09.519
graphs are a lot less helpful this is a
00:15:09.519 --> 00:15:11.120
highly important equation in its own
00:15:11.120 --> 00:15:13.040
right it's a very powerful concept when
00:15:13.040 --> 00:15:14.480
the rate of change of a value
00:15:14.480 --> 00:15:16.880
is proportional to the value itself this
00:15:16.880 --> 00:15:18.480
is the equation governing things like
00:15:18.480 --> 00:15:19.839
compound interest
00:15:19.839 --> 00:15:22.079
or the early stages of population growth
00:15:22.079 --> 00:15:23.920
before the effects of limited resources
00:15:23.920 --> 00:15:25.199
kick in
00:15:25.199 --> 00:15:27.199
or the early stages of an epidemic while
00:15:27.199 --> 00:15:31.680
most of the population is susceptible
00:15:31.680 --> 00:15:33.519
calculus students all learn about how
00:15:33.519 --> 00:15:35.920
the derivative of e to the r t
00:15:35.920 --> 00:15:39.040
is r times itself in other words
00:15:39.040 --> 00:15:41.199
this self-reinforcing growth phenomenon
00:15:41.199 --> 00:15:43.680
is the same thing as exponential growth
00:15:43.680 --> 00:15:48.800
and e to the rt solves this equation
00:15:48.800 --> 00:15:50.320
actually a better way to think about it
00:15:50.320 --> 00:15:51.759
is that there are many different
00:15:51.759 --> 00:15:53.279
solutions to this equation
00:15:53.279 --> 00:15:55.519
one for each initial condition something
00:15:55.519 --> 00:15:57.360
like an initial investment size or an
00:15:57.360 --> 00:15:58.560
initial population
00:15:58.560 --> 00:16:01.360
which we'll just call x naught notice by
00:16:01.360 --> 00:16:01.759
the way
00:16:01.759 --> 00:16:04.079
how the higher the value for x naught
00:16:04.079 --> 00:16:05.680
the higher the initial slope of the
00:16:05.680 --> 00:16:07.360
resulting solution
00:16:07.360 --> 00:16:09.120
which should make complete sense given
00:16:09.120 --> 00:16:11.440
the equation
00:16:11.440 --> 00:16:13.519
the function e to the rt is just a
00:16:13.519 --> 00:16:15.519
solution when the initial condition
00:16:15.519 --> 00:16:18.720
is one but if you multiply by any other
00:16:18.720 --> 00:16:19.920
initial condition
00:16:19.920 --> 00:16:21.519
you get a new function which still
00:16:21.519 --> 00:16:23.360
satisfies this property it still has a
00:16:23.360 --> 00:16:24.560
derivative which is r
00:16:24.560 --> 00:16:27.440
times itself but this time it starts at
00:16:27.440 --> 00:16:28.160
x naught
00:16:28.160 --> 00:16:31.120
since e to the 0 is 1. this is worth
00:16:31.120 --> 00:16:32.560
highlighting before we generalize to
00:16:32.560 --> 00:16:33.680
more dimensions
00:16:33.680 --> 00:16:35.360
do not think of the exponential part as
00:16:35.360 --> 00:16:37.600
being a solution in and of itself
00:16:37.600 --> 00:16:39.680
think of it as something that acts on an
00:16:39.680 --> 00:16:40.720
initial condition
00:16:40.720 --> 00:16:46.320
in order to give a solution
00:16:46.320 --> 00:16:48.399
you see up in the two-dimensional case
00:16:48.399 --> 00:16:50.240
when we have a changing vector whose
00:16:50.240 --> 00:16:51.920
rate of change is constrained to be some
00:16:51.920 --> 00:16:54.079
matrix times itself
00:16:54.079 --> 00:16:56.399
what the solution looks like is also an
00:16:56.399 --> 00:16:57.519
exponential term
00:16:57.519 --> 00:16:59.839
acting on a given initial condition but
00:16:59.839 --> 00:17:01.759
the exponential part in that case
00:17:01.759 --> 00:17:03.600
will produce a matrix that changes with
00:17:03.600 --> 00:17:05.439
time and the initial condition
00:17:05.439 --> 00:17:08.000
is a vector in fact you should think of
00:17:08.000 --> 00:17:10.319
the definition of matrix exponentiation
00:17:10.319 --> 00:17:12.480
as being heavily motivated by making
00:17:12.480 --> 00:17:14.799
sure that this fact is true
00:17:14.799 --> 00:17:16.400
for example if we look back at the
00:17:16.400 --> 00:17:18.240
system that popped up with romeo and
00:17:18.240 --> 00:17:19.120
juliet
00:17:19.120 --> 00:17:21.360
the claim now is that solutions look
00:17:21.360 --> 00:17:23.039
like e raised to this
00:17:23.039 --> 00:17:26.799
0 negative 1 1 0 matrix all times time
00:17:26.799 --> 00:17:29.679
multiplied by some initial condition but
00:17:29.679 --> 00:17:31.120
we've already seen the solution in this
00:17:31.120 --> 00:17:32.799
case we know it looks like a rotation
00:17:32.799 --> 00:17:33.360
matrix
00:17:33.360 --> 00:17:35.440
times the initial condition so let's
00:17:35.440 --> 00:17:37.200
take a moment to roll up our sleeves and
00:17:37.200 --> 00:17:38.960
compute the exponential term
00:17:38.960 --> 00:17:40.480
using the definition that i mentioned at
00:17:40.480 --> 00:17:43.039
the start and see if it lines up
00:17:43.039 --> 00:17:45.039
remember writing e to the power of a
00:17:45.039 --> 00:17:46.799
matrix is a shorthand
00:17:46.799 --> 00:17:48.720
a shorthand for plugging it in to this
00:17:48.720 --> 00:17:50.320
long infinite polynomial
00:17:50.320 --> 00:17:53.360
the taylor series for e to the x
00:17:53.360 --> 00:17:55.120
i know it might seem pretty complicated
00:17:55.120 --> 00:17:56.720
to do this but trust me
00:17:56.720 --> 00:17:58.559
it's very satisfying how this particular
00:17:58.559 --> 00:18:00.000
one turns out
00:18:00.000 --> 00:18:01.919
if you actually sit down and you compute
00:18:01.919 --> 00:18:04.160
successive powers of this matrix
00:18:04.160 --> 00:18:05.840
what you'd notice is that they fall into
00:18:05.840 --> 00:18:15.680
a cycling pattern every four iterations
00:18:15.680 --> 00:18:21.970
[Music]
00:18:21.970 --> 00:18:27.360
[Music]
00:18:27.360 --> 00:18:28.799
this should make sense given that we
00:18:28.799 --> 00:18:31.280
know it's a 90 degree rotation matrix
00:18:31.280 --> 00:18:33.200
so when you add together all infinitely
00:18:33.200 --> 00:18:35.360
many matrices term by term
00:18:35.360 --> 00:18:37.200
each term in the result looks like a
00:18:37.200 --> 00:18:38.880
polynomial in t
00:18:38.880 --> 00:18:40.840
with some nice cycling pattern in its
00:18:40.840 --> 00:18:42.880
coefficients all of them scaled by the
00:18:42.880 --> 00:18:45.520
relevant factorial term
00:18:45.520 --> 00:18:47.200
those of you who are savvy with taylor
00:18:47.200 --> 00:18:49.120
series might be able to recognize
00:18:49.120 --> 00:18:51.120
that each one of these components is the
00:18:51.120 --> 00:18:52.559
taylor series for either
00:18:52.559 --> 00:18:54.880
sine or cosine though in that top right
00:18:54.880 --> 00:18:56.559
corner's case it's actually negative
00:18:56.559 --> 00:18:58.799
sign
00:18:58.799 --> 00:19:00.880
so what we get from the computation is
00:19:00.880 --> 00:19:02.799
exactly the rotation matrix we had from
00:19:02.799 --> 00:19:07.280
before
00:19:07.280 --> 00:19:09.600
to me this is extremely beautiful we
00:19:09.600 --> 00:19:10.320
have two
00:19:10.320 --> 00:19:12.000
completely different ways of reasoning
00:19:12.000 --> 00:19:13.360
about the same system
00:19:13.360 --> 00:19:15.440
and they give us the same answer i mean
00:19:15.440 --> 00:19:16.960
it's reassuring that they do
00:19:16.960 --> 00:19:18.880
but it is wild just how different the
00:19:18.880 --> 00:19:20.320
mode of thought is when you're chugging
00:19:20.320 --> 00:19:21.600
through this polynomial
00:19:21.600 --> 00:19:22.799
versus when you're geometrically
00:19:22.799 --> 00:19:24.480
reasoning about what a velocity
00:19:24.480 --> 00:19:27.520
perpendicular to a position must imply
00:19:27.520 --> 00:19:29.360
hopefully the fact that these line up
00:19:29.360 --> 00:19:30.720
inspires a little confidence in the
00:19:30.720 --> 00:19:32.240
claim that matrix exponents
00:19:32.240 --> 00:19:35.360
really do solve systems like this this
00:19:35.360 --> 00:19:36.960
explains the computation we saw at the
00:19:36.960 --> 00:19:38.559
start by the way with the matrix that
00:19:38.559 --> 00:19:38.880
had
00:19:38.880 --> 00:19:41.280
negative pi and pi off the diagonals
00:19:41.280 --> 00:19:43.360
producing the negative identity
00:19:43.360 --> 00:19:45.760
this expression is exponentiating a 90
00:19:45.760 --> 00:19:47.200
degree rotation matrix
00:19:47.200 --> 00:19:49.280
times pi which is another way to
00:19:49.280 --> 00:19:51.440
describe what the romeo juliet setup
00:19:51.440 --> 00:19:52.559
does after pi
00:19:52.559 --> 00:19:55.120
units of time as we now know that has
00:19:55.120 --> 00:19:56.720
the effect of rotating everything
00:19:56.720 --> 00:19:59.600
180 degrees in this state space which is
00:19:59.600 --> 00:20:02.880
the same as multiplying by negative one
00:20:02.880 --> 00:20:04.480
also for any of you familiar with
00:20:04.480 --> 00:20:06.159
imaginary number exponents
00:20:06.159 --> 00:20:07.840
this particular example is probably
00:20:07.840 --> 00:20:09.200
ringing a ton of bells
00:20:09.200 --> 00:20:12.240
it is 100 analogous in fact we could
00:20:12.240 --> 00:20:13.840
have framed the entire example where
00:20:13.840 --> 00:20:15.360
romeo and juliet's feelings
00:20:15.360 --> 00:20:17.600
were packaged into a complex number and
00:20:17.600 --> 00:20:19.120
the rate of change of that complex
00:20:19.120 --> 00:20:19.600
number
00:20:19.600 --> 00:20:21.840
would have been i times itself since
00:20:21.840 --> 00:20:23.120
multiplication by i
00:20:23.120 --> 00:20:25.840
also acts like a 90 degree rotation the
00:20:25.840 --> 00:20:27.679
same exact line of reasoning both
00:20:27.679 --> 00:20:29.280
analytic and geometric
00:20:29.280 --> 00:20:31.280
would have led to this whole idea that e
00:20:31.280 --> 00:20:32.559
to the power i t
00:20:32.559 --> 00:20:34.960
describes rotation these are actually
00:20:34.960 --> 00:20:36.640
two of many different examples
00:20:36.640 --> 00:20:37.919
throughout math and physics
00:20:37.919 --> 00:20:39.520
when you find yourself exponentiating
00:20:39.520 --> 00:20:41.600
some object which acts as a 90 degree
00:20:41.600 --> 00:20:42.400
rotation
00:20:42.400 --> 00:20:45.200
times time it shows up with quaternions
00:20:45.200 --> 00:20:47.120
or many of the matrices that pop up in
00:20:47.120 --> 00:20:48.640
quantum mechanics
00:20:48.640 --> 00:20:50.320
in all of these cases we have this
00:20:50.320 --> 00:20:51.840
really neat general idea
00:20:51.840 --> 00:20:53.360
that if you take some operation that
00:20:53.360 --> 00:20:55.840
rotates 90 degrees in some plane
00:20:55.840 --> 00:20:57.280
often it's a plane in some high
00:20:57.280 --> 00:20:58.480
dimensional space that we can't
00:20:58.480 --> 00:20:59.520
visualize
00:20:59.520 --> 00:21:01.600
then what we get by exponentiating that
00:21:01.600 --> 00:21:03.360
operation times time
00:21:03.360 --> 00:21:05.200
is something that generates all other
00:21:05.200 --> 00:21:09.360
rotations in that same plane
00:21:09.360 --> 00:21:11.120
one of the more complicated variations
00:21:11.120 --> 00:21:12.799
on this same theme is schrodinger's
00:21:12.799 --> 00:21:13.600
equation
00:21:13.600 --> 00:21:15.440
it's not just that this has the
00:21:15.440 --> 00:21:17.280
derivative of a state equals some matrix
00:21:17.280 --> 00:21:18.320
times that state
00:21:18.320 --> 00:21:20.960
form the nature of the relevant matrix
00:21:20.960 --> 00:21:21.600
here
00:21:21.600 --> 00:21:23.679
is such that the equation also describes
00:21:23.679 --> 00:21:25.039
a kind of rotation
00:21:25.039 --> 00:21:26.400
though in many applications of
00:21:26.400 --> 00:21:27.600
schrodinger's equation it'll be a
00:21:27.600 --> 00:21:30.240
rotation in a kind of function space
00:21:30.240 --> 00:21:31.600
it's a little more involved though
00:21:31.600 --> 00:21:33.360
because typically there's a combination
00:21:33.360 --> 00:21:35.120
of many different rotations
00:21:35.120 --> 00:21:36.880
it takes time to really dig into this
00:21:36.880 --> 00:21:38.480
equation and i would love to do that in
00:21:38.480 --> 00:21:39.679
a later chapter
00:21:39.679 --> 00:21:41.840
but right now i cannot help but at least
00:21:41.840 --> 00:21:43.200
allude to the fact
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that this imaginary unit i that sits so
00:21:45.760 --> 00:21:47.600
impishly in such a fundamental equation
00:21:47.600 --> 00:21:48.960
for all of the universe
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is playing basically the same role as
00:21:51.280 --> 00:21:54.159
the matrix from our romeo juliet example
00:21:54.159 --> 00:21:56.320
what this i communicates is that the
00:21:56.320 --> 00:21:58.400
rate of change of a certain state
00:21:58.400 --> 00:22:00.880
is in a sense perpendicular to that
00:22:00.880 --> 00:22:01.679
state
00:22:01.679 --> 00:22:03.679
and hence that the way things have to
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evolve over time
00:22:04.880 --> 00:22:10.960
will involve a kind of oscillation
00:22:10.960 --> 00:22:13.120
but matrix exponentiation can do so much
00:22:13.120 --> 00:22:14.880
more than just rotation
00:22:14.880 --> 00:22:16.720
you can always visualize these sorts of
00:22:16.720 --> 00:22:17.919
differential equations
00:22:17.919 --> 00:22:21.200
using a vector field the idea is that
00:22:21.200 --> 00:22:23.120
this equation tells us the velocity of a
00:22:23.120 --> 00:22:23.600
state
00:22:23.600 --> 00:22:26.159
is entirely determined by its position
00:22:26.159 --> 00:22:27.760
so what we do is go to every point in
00:22:27.760 --> 00:22:28.400
the space
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and draw a little vector indicating what
00:22:30.640 --> 00:22:32.880
the velocity of a state must be
00:22:32.880 --> 00:22:35.600
if it passes through that point for our
00:22:35.600 --> 00:22:36.720
type of equation
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this means that we go to each point v in
00:22:38.480 --> 00:22:40.400
space and we attach the vector
00:22:40.400 --> 00:22:54.000
m times v
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to intuitively understand how any given
00:22:56.080 --> 00:22:57.679
initial condition will evolve
00:22:57.679 --> 00:23:00.000
you let it flow along this field with a
00:23:00.000 --> 00:23:01.440
velocity always matching
00:23:01.440 --> 00:23:03.120
whatever vector it's sitting on at any
00:23:03.120 --> 00:23:05.600
given point in time
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so if the claim is that solutions to
00:23:07.360 --> 00:23:08.880
this equation look like
00:23:08.880 --> 00:23:12.000
e to the mt times some initial condition
00:23:12.000 --> 00:23:13.600
it means you can visualize what the
00:23:13.600 --> 00:23:15.520
matrix e to the mt does
00:23:15.520 --> 00:23:17.280
by letting every possible initial
00:23:17.280 --> 00:23:19.919
condition flow along this field for t
00:23:19.919 --> 00:23:24.799
units of time
00:23:24.799 --> 00:23:27.520
the transition from start to finish is
00:23:27.520 --> 00:23:29.840
described by whatever matrix pops out
00:23:29.840 --> 00:23:30.880
from the computation
00:23:30.880 --> 00:23:34.480
for e to the mt in our main example with
00:23:34.480 --> 00:23:36.400
the 90 degree rotation matrix
00:23:36.400 --> 00:23:38.720
the vector field looks like this and as
00:23:38.720 --> 00:23:40.320
we saw e to the mt
00:23:40.320 --> 00:23:42.400
describes rotation in that case which
00:23:42.400 --> 00:23:45.600
lines up with flow along this field
00:23:45.600 --> 00:23:47.360
as another example the more
00:23:47.360 --> 00:23:49.279
shakespearean romeo and juliet
00:23:49.279 --> 00:23:50.640
might have equations that look a little
00:23:50.640 --> 00:23:52.799
more like this where juliet's rule is
00:23:52.799 --> 00:23:54.320
symmetric with romeos
00:23:54.320 --> 00:23:55.919
and both of them are inclined to get
00:23:55.919 --> 00:23:57.440
carried away in response to one
00:23:57.440 --> 00:23:59.200
another's feelings
00:23:59.200 --> 00:24:00.720
again the way the vector field you're
00:24:00.720 --> 00:24:02.720
looking at has been defined is to go to
00:24:02.720 --> 00:24:03.600
each point v
00:24:03.600 --> 00:24:07.039
in space and attach the vector m times v
00:24:07.039 --> 00:24:08.960
this is the pictorial way of saying that
00:24:08.960 --> 00:24:10.640
the rate of change of a state
00:24:10.640 --> 00:24:14.159
must always equal m times itself
00:24:14.159 --> 00:24:16.159
but for this example flow along the
00:24:16.159 --> 00:24:17.919
field looks a lot different from how it
00:24:17.919 --> 00:24:19.039
did before
00:24:19.039 --> 00:24:21.200
if romeo and juliet start off anywhere
00:24:21.200 --> 00:24:23.440
in this upper right half of the plane
00:24:23.440 --> 00:24:24.960
their feelings will feed off of each
00:24:24.960 --> 00:24:26.559
other and they both tend towards
00:24:26.559 --> 00:24:27.700
infinity
00:24:27.700 --> 00:24:30.720
[Music]
00:24:30.720 --> 00:24:31.919
if they're in the other half of the
00:24:31.919 --> 00:24:34.080
plane well let's just say that they stay
00:24:34.080 --> 00:24:35.919
more true to their capulet and montague
00:24:35.919 --> 00:24:37.760
family traditions
00:24:37.760 --> 00:24:39.679
so even before you try calculating the
00:24:39.679 --> 00:24:42.000
exponential of this particular matrix
00:24:42.000 --> 00:24:43.679
you can already have an intuitive sense
00:24:43.679 --> 00:24:46.080
for what the answer should look like the
00:24:46.080 --> 00:24:47.120
resulting matrix
00:24:47.120 --> 00:24:48.880
should describe the transition from time
00:24:48.880 --> 00:24:50.559
0 to time t
00:24:50.559 --> 00:24:52.640
which if you look at the field seems to
00:24:52.640 --> 00:24:54.000
indicate that it will squish
00:24:54.000 --> 00:24:56.080
along one diagonal while stretching
00:24:56.080 --> 00:24:57.039
along another
00:24:57.039 --> 00:25:00.640
getting more extreme as t gets larger
00:25:00.640 --> 00:25:02.400
of course all of this is presuming that
00:25:02.400 --> 00:25:05.120
e to the mt times an initial condition
00:25:05.120 --> 00:25:07.679
actually solves these systems this is
00:25:07.679 --> 00:25:09.200
one of those facts that's easiest to
00:25:09.200 --> 00:25:10.559
believe when you just work it out
00:25:10.559 --> 00:25:12.320
yourself
00:25:12.320 --> 00:25:13.679
but i'll run through a quick rough
00:25:13.679 --> 00:25:15.919
sketch
00:25:15.919 --> 00:25:17.360
write out the full polynomial that
00:25:17.360 --> 00:25:19.120
defines e to the mt
00:25:19.120 --> 00:25:21.120
and multiply by some initial condition
00:25:21.120 --> 00:25:26.720
vector on the right
00:25:26.720 --> 00:25:28.159
and then take the derivative of this
00:25:28.159 --> 00:25:30.000
with respect to t
00:25:30.000 --> 00:25:32.080
because the matrix m is a constant this
00:25:32.080 --> 00:25:33.679
just means applying the power rule to
00:25:33.679 --> 00:25:43.520
each one of the terms
00:25:43.520 --> 00:25:45.120
and that power rule really nicely
00:25:45.120 --> 00:25:48.650
cancels out with the factorial terms
00:25:48.650 --> 00:25:53.440
[Music]
00:25:53.440 --> 00:25:54.960
so what we're left with is an expression
00:25:54.960 --> 00:25:56.960
that looks almost identical to what we
00:25:56.960 --> 00:25:58.000
had before
00:25:58.000 --> 00:25:59.840
except that each term has an extra m
00:25:59.840 --> 00:26:01.039
hanging onto it
00:26:01.039 --> 00:26:03.520
but this can be factored out to the left
00:26:03.520 --> 00:26:05.679
so the derivative of the expression
00:26:05.679 --> 00:26:08.880
is m times the original expression and
00:26:08.880 --> 00:26:11.200
hence it solves the equation
00:26:11.200 --> 00:26:12.960
this actually sweeps under the rug some
00:26:12.960 --> 00:26:14.640
details required for bigger
00:26:14.640 --> 00:26:16.159
mostly centered around the question of
00:26:16.159 --> 00:26:17.440
whether or not this thing actually
00:26:17.440 --> 00:26:18.320
converges
00:26:18.320 --> 00:26:21.039
but it does give the main idea in the
00:26:21.039 --> 00:26:22.640
next chapter i would like to talk more
00:26:22.640 --> 00:26:24.400
about the properties that this operation
00:26:24.400 --> 00:26:25.039
has
00:26:25.039 --> 00:26:26.559
most notably its relationship with
00:26:26.559 --> 00:26:28.480
eigenvectors and eigenvalues
00:26:28.480 --> 00:26:30.400
which leads us to more concrete ways of
00:26:30.400 --> 00:26:32.000
thinking about how you actually carry
00:26:32.000 --> 00:26:33.200
out this computation
00:26:33.200 --> 00:26:36.320
which otherwise seems insane also
00:26:36.320 --> 00:26:37.919
time permitting it might be fun to talk
00:26:37.919 --> 00:26:39.440
about what it means to raise e to the
00:26:39.440 --> 00:26:42.980
power of the derivative
00:26:42.980 --> 00:26:50.840
[Music]
00:26:50.840 --> 00:26:55.510
operator
00:26:55.510 --> 00:27:06.720
[Music]
00:27:06.720 --> 00:27:08.799
you