1
00:00:08.22 --> 00:00:12
-- and lift-off on differential
equations.
2
00:00:12 --> 00:00:19
So, this section is about how
to solve a system of first
3
00:00:19 --> 00:00:25
order, first derivative,
constant coefficient linear
4
00:00:25 --> 00:00:26
equations.
5
00:00:26 --> 00:00:32
And if we do it right,
it turns directly into linear
6
00:00:32 --> 00:00:34.5
algebra.
7
00:00:34.5 --> 00:00:39
The key idea is the solutions
to constant coefficient linear
8
00:00:39 --> 00:00:41
equations are exponentials.
9
00:00:41 --> 00:00:45.57
So if you look for an
exponential, then all you have
10
00:00:45.57 --> 00:00:50
to find is what's up there in
the exponent and what multiplies
11
00:00:50 --> 00:00:54
the exponential and that's the
linear algebra.
12
00:00:54 --> 00:00:58
So -- and the result -- one
thing we will fine --
13
00:00:58 --> 00:01:03
it's completely parallel to
powers of a matrix.
14
00:01:03 --> 00:01:08
So the last lecture was about
how would you compute A to the K
15
00:01:08 --> 00:01:10
or A to the 100?
16
00:01:10 --> 00:01:13
How do you compute high powers
of a matrix?
17
00:01:13 --> 00:01:18
Now it's not powers anymore,
but it's exponentials.
18
00:01:18 --> 00:01:23
That's the natural thing for
differential equation.
19
00:01:23 --> 00:01:23
Okay.
20
00:01:23 --> 00:01:26
But can I begin with an
example?
21
00:01:26 --> 00:01:28
And I'll just go through the
mechanics.
22
00:01:28 --> 00:01:32
How would I solve the
differential -- two differential
23
00:01:32 --> 00:01:33
equations?
24
00:01:33 --> 00:01:38
So I'm going to make it -- I'll
have a two by two matrix and the
25
00:01:38 --> 00:01:44
coefficients are minus one two,
one minus two and I'd better
26
00:01:44 --> 00:01:46
give you some initial condition.
27
00:01:46 --> 00:01:51.54
So suppose it starts u at times
zero -- this is u1,
28
00:01:51.54 --> 00:01:56
u2 -- let it -- let it --
suppose everything is in u1 at
29
00:01:56 --> 00:01:57
times zero.
30
00:01:57 --> 00:02:01
So -- at -- at the start,
it's all in u1.
31
00:02:01 --> 00:02:05
But what happens as time goes
on,
32
00:02:05 --> 00:02:10
du2/dt will -- will be
positive, because of that u1
33
00:02:10 --> 00:02:16
term, so flow will move into the
u2 component and it will go out
34
00:02:16 --> 00:02:18
of the u1 component.
35
00:02:18 --> 00:02:24
So we'll just follow that
movement as time goes forward by
36
00:02:24 --> 00:02:31
looking at the eigenvalues and
eigenvectors of that matrix.
37
00:02:31 --> 00:02:33.48
That's a first job.
38
00:02:33.48 --> 00:02:38.72
Before you do anything else,
find the -- find the matrix and
39
00:02:38.72 --> 00:02:41
its eigenvalues and
eigenvectors.
40
00:02:41 --> 00:02:43
So let me do that.
41
00:02:43 --> 00:02:43
Okay.
42
00:02:43 --> 00:02:45.47
So here's our matrix.
43
00:02:45.47 --> 00:02:50
Maybe you can tell me right
away what -- what are the
44
00:02:50 --> 00:02:53
eigenvalues and -- eigenvalues
anyway.
45
00:02:53 --> 00:02:56
And then we can check.
46
00:02:56 --> 00:03:00.27
But can you spot any of the
eigenvalues of that matrix?
47
00:03:00.27 --> 00:03:02
We're looking for two
eigenvalues.
48
00:03:02 --> 00:03:05
Do you see -- I mean,
if I just wrote that matrix
49
00:03:05 --> 00:03:08
down, what -- what do you notice
about it?
50
00:03:08 --> 00:03:10
It's singular,
right.
51
00:03:10 --> 00:03:13
That -- that's a singular
matrix.
52
00:03:13 --> 00:03:18
That tells me right away that
one of the eigenvalues is lambda
53
00:03:18 --> 00:03:19
equals zero.
54
00:03:19 --> 00:03:23
I can -- that's a singular
matrix, the second column is
55
00:03:23 --> 00:03:28
minus two times the first
column, the determinant is zero,
56
00:03:28 --> 00:03:32
it's -- it's singular,
so zero is an eigenvalue and
57
00:03:32 --> 00:03:36.62
the other eigenvalue will
be -- from the trace.
58
00:03:36.62 --> 00:03:39
I look at the trace,
the sum down the diagonal is
59
00:03:39 --> 00:03:40
minus three.
60
00:03:40 --> 00:03:43
That has to agree with the sum
of the eigenvalue,
61
00:03:43 --> 00:03:47.15
so that second eigenvalue
better be minus three.
62
00:03:47.15 --> 00:03:51
I could, of course -- I could
compute -- why don't I over
63
00:03:51 --> 00:03:56
here -- compute the determinant
of A minus lambda I,
64
00:03:56 --> 00:04:01
the determinant of this minus
one minus lambda two one minus
65
00:04:01 --> 00:04:03
two minus lambda matrix.
66
00:04:03 --> 00:04:05
But we know what's coming.
67
00:04:05 --> 00:04:09
When I do that multiplication,
I get a lambda squared.
68
00:04:09 --> 00:04:13
I get a two lambda and a one
lambda,
69
00:04:13 --> 00:04:15
that's a three lambda.
70
00:04:15 --> 00:04:19
And then -- now I'm going to
get the determinant,
71
00:04:19 --> 00:04:22
which is two minus two which is
zero.
72
00:04:22 --> 00:04:27
So there's my characteristic
polynomial, this determinant.
73
00:04:27 --> 00:04:32
And of course I factor that
into lambda times lambda plus
74
00:04:32 --> 00:04:36
three and I get the two
eigenvalues
75
00:04:36 --> 00:04:37
that we saw coming.
76
00:04:37 --> 00:04:39
What else do I need?
77
00:04:39 --> 00:04:40
The eigenvectors.
78
00:04:40 --> 00:04:45
So before I even think about
the differential equation or
79
00:04:45 --> 00:04:49
what -- how to solve it,
let me find the eigenvectors
80
00:04:49 --> 00:04:50
for this matrix.
81
00:04:50 --> 00:04:51
Okay.
82
00:04:51 --> 00:04:55
So take lambda equals zero --
so that -- that's the first
83
00:04:55 --> 00:04:57
eigenvalue.
84
00:04:57 --> 00:05:02
Lambda one equals zero and the
second eigenvalue will be lambda
85
00:05:02 --> 00:05:04
two equals minus three.
86
00:05:04 --> 00:05:09
By the way, I -- I already know
something important about this.
87
00:05:09 --> 00:05:13.02
The eigenvalues are telling me
something.
88
00:05:13.02 --> 00:05:18
You'll see how it comes out,
but let me point to -- these
89
00:05:18 --> 00:05:22
numbers are -- this eigenvalue,
a negative eigenvalue,
90
00:05:22 --> 00:05:24
is going to disappear.
91
00:05:24 --> 00:05:29
There's going to be an e to the
minus three t in the answer.
92
00:05:29 --> 00:05:34
That e to the minus three t as
times goes on is going to be
93
00:05:34 --> 00:05:35
very, very small.
94
00:05:35 --> 00:05:39
The other part of the answer
will
95
00:05:39 --> 00:05:41
involve an e to the zero t.
96
00:05:41 --> 00:05:45
But e to the zero t is one and
that's a constant.
97
00:05:45 --> 00:05:49
So I'm expecting that this
solution'll have two parts,
98
00:05:49 --> 00:05:54
an e to the zero t part and an
e to the minus three t part,
99
00:05:54 --> 00:05:59
and that -- and as time goes
on, the second part'll disappear
100
00:05:59 --> 00:06:03.14
and the
first part will be a steady
101
00:06:03.14 --> 00:06:03
state.
102
00:06:03 --> 00:06:04
It won't move.
103
00:06:04 --> 00:06:08
It will be -- at the end of --
as t approaches infinity,
104
00:06:08 --> 00:06:13
this part disappears and this
is the -- the e to the zero t
105
00:06:13 --> 00:06:14
part is what I get.
106
00:06:14 --> 00:06:18
And I'm very interested in
these steady states,
107
00:06:18 --> 00:06:22
so that's -- I get a steady
state when I have a zero
108
00:06:22 --> 00:06:22
eigenvalue.
109
00:06:22 --> 00:06:23
Okay.
110
00:06:23 --> 00:06:25
What about those eigenvectors?
111
00:06:25 --> 00:06:28
So what's the eigenvector that
goes with eigenvalue zero?
112
00:06:28 --> 00:06:29.23
Okay.
113
00:06:29.23 --> 00:06:33
The matrix is singular as it
is, the eigenvector is -- is the
114
00:06:33 --> 00:06:36
guy in the null space,
so what vector is in the null
115
00:06:36 --> 00:06:39
space
of that matrix?
116
00:06:39 --> 00:06:40
Let's see.
117
00:06:40 --> 00:06:46
I guess I probably give the
free variable the value one and
118
00:06:46 --> 00:06:52
I realize that if I want to get
zero I need a two up here.
119
00:06:52 --> 00:06:52
Okay?
120
00:06:52 --> 00:06:54.54
So Ax1 is zero x1.
121
00:06:54.54 --> 00:06:56
A x1 is zero x1.
122
00:06:56 --> 00:06:56
Fine.
123
00:06:56 --> 00:06:57
Okay.
124
00:06:57 --> 00:07:01
What about the other
eigenvalue?
125
00:07:01 --> 00:07:03
Lambda two is minus three.
126
00:07:03 --> 00:07:04
Okay.
127
00:07:04 --> 00:07:06
How do I get the other
eigenvalue, then?
128
00:07:06 --> 00:07:09.69
For the moment -- can I
mentally do it?
129
00:07:09.69 --> 00:07:12
I subtract minus three along
the diagonal,
130
00:07:12 --> 00:07:16
which means I add three -- can
I -- I'll just do it with an
131
00:07:16 --> 00:07:19
erase -- erase for the moment.
132
00:07:19 --> 00:07:23
So I'm going to add three
to the diagonal.
133
00:07:23 --> 00:07:28
So this minus one will become a
two and -- I'll make it in big
134
00:07:28 --> 00:07:32
loopy letters -- and when I add
three to this guy,
135
00:07:32 --> 00:07:37
the minus two becomes -- well,
I can't make one very loopy,
136
00:07:37 --> 00:07:38
but how's that?
137
00:07:38 --> 00:07:38
Okay.
138
00:07:38 --> 00:07:42
Now that's A minus three I -- A
plus three I,
139
00:07:42 --> 00:07:43
sorry.
140
00:07:43 --> 00:07:45
That's A plus three I.
141
00:07:45 --> 00:07:48.1
It's supposed to be singular,
right?
142
00:07:48.1 --> 00:07:52
I-- if things -- if I did it
right, this matrix should be
143
00:07:52 --> 00:07:55
singular and the x2,
the eigenvector should be in
144
00:07:55 --> 00:07:56
its null space.
145
00:07:56 --> 00:07:57
Okay.
146
00:07:57 --> 00:08:01.2
What do I get for
the null space of this?
147
00:08:01.2 --> 00:08:04
Maybe minus one one,
or one minus one.
148
00:08:04 --> 00:08:05.89
Doesn't matter.
149
00:08:05.89 --> 00:08:08
Those are both perfectly good.
150
00:08:08 --> 00:08:09
Right?
151
00:08:09 --> 00:08:12
Because that's in the null
space of this.
152
00:08:12 --> 00:08:17
Now I'll -- because A times
that vector is three times that
153
00:08:17 --> 00:08:18
vector.
154
00:08:18 --> 00:08:21
Ax2 is minus three
x2.
155
00:08:21 --> 00:08:22
Good.
156
00:08:22 --> 00:08:22
Okay.
157
00:08:22 --> 00:08:26
Can I get A again so we see
that correctly?
158
00:08:26 --> 00:08:30
That was a minus one and that
was a minus two.
159
00:08:30 --> 00:08:31
Good.
160
00:08:31 --> 00:08:31
Okay.
161
00:08:31 --> 00:08:36
That -- that's the first job.
eigenvalues and eigenvectors.
162
00:08:36 --> 00:08:42
And already the eigenvalues are
telling me the most important
163
00:08:42 --> 00:08:46
information about the answer.
164
00:08:46 --> 00:08:49
But now, what is the answer?
165
00:08:49 --> 00:08:54
The answer is -- the solution
will be U of T -- okay.
166
00:08:54 --> 00:08:59
Now, wh- now I use those
eigenvalues and eigenvectors.
167
00:08:59 --> 00:09:04
The solution is some -- there
are two eigenvalues.
168
00:09:04 --> 00:09:09
So I -- it -- so there're going
to be two special
169
00:09:09 --> 00:09:10
solutions here.
170
00:09:10 --> 00:09:13
Two pure exponential solutions.
171
00:09:13 --> 00:09:19
The first one is going to be
either the lambda one tx1 and
172
00:09:19 --> 00:09:24
the -- so that solves the
equation, and so does this one.
173
00:09:24 --> 00:09:29
They both are solutions to the
differential equation.
174
00:09:29 --> 00:09:32
That's the general solution.
175
00:09:32 --> 00:09:37
The general solution is a
combination of that pure
176
00:09:37 --> 00:09:41
exponential solution and that
pure exponential solution.
177
00:09:41 --> 00:09:46
Can I just see that those guys
do solve the equation?
178
00:09:46 --> 00:09:50.48
So let me just check -- check
on this one, for example.
179
00:09:50.48 --> 00:09:50
Check.
180
00:09:50 --> 00:09:55
I -- I want to check that the
-- my equation -- let's
181
00:09:55 --> 00:09:59
remember,
the equation -- du/dt is Au.
182
00:09:59 --> 00:10:05
I plug in e to the lambda one t
x1 and let's just see that the
183
00:10:05 --> 00:10:07
equation's okay.
184
00:10:07 --> 00:10:11
I believe this is a solution to
that equation.
185
00:10:11 --> 00:10:13
So just plug it in.
186
00:10:13 --> 00:10:18
On the left-hand side,
I take the time derivative --
187
00:10:18 --> 00:10:23.25
so the left-hand side will be
lambda
188
00:10:23.25 --> 00:10:26
one, e to the lambda one t x1,
right?
189
00:10:26 --> 00:10:30
The time derivative -- this is
the term that depends on time,
190
00:10:30 --> 00:10:35
it's just ordinary exponential,
its derivative brings down a
191
00:10:35 --> 00:10:36
lambda one.
192
00:10:36 --> 00:10:40
On the other side of the
equation it's A times this
193
00:10:40 --> 00:10:40
thing.
194
00:10:40 --> 00:10:43
A times either the lambda one t
x
195
00:10:43 --> 00:10:46
one, and does that check out?
196
00:10:46 --> 00:10:49
Do we have equality there?
197
00:10:49 --> 00:10:54
Yes, because either the lambda
one t appears on both sides and
198
00:10:54 --> 00:10:59
the other one is Ax1 equal
lambda one x1 -- check.
199
00:10:59 --> 00:11:03
Do you -- so,
the -- we've come to the first
200
00:11:03 --> 00:11:05
point to remember.
201
00:11:05 --> 00:11:08
These pure solutions.
202
00:11:08 --> 00:11:14
Those pure solutions are the --
those pure exponentials are the
203
00:11:14 --> 00:11:19
differential equations analogue
of -- last time we had pure
204
00:11:19 --> 00:11:20
powers.
205
00:11:20 --> 00:11:25
Last time -- so -- so last
time, the analog was lambda --
206
00:11:25 --> 00:11:30
lambda one to the K-th power x1,
some amount of that,
207
00:11:30 --> 00:11:34
plus some amount of lambda two
to
208
00:11:34 --> 00:11:36
the K-th power x2.
209
00:11:36 --> 00:11:39
That was our formula from last
time.
210
00:11:39 --> 00:11:45
I put it up just to -- so your
eye compares those two formulas.
211
00:11:45 --> 00:11:50
Powers of lambda in the -- in
the difference equation -- that
212
00:11:50 --> 00:11:55
-- this was in the -- this was
for the equation uk plus one
213
00:11:55 --> 00:11:57
equals A uk.
214
00:11:57 --> 00:12:01
That was for the finite step --
stepping by one.
215
00:12:01 --> 00:12:04
And we got powers,
now this is the one we're
216
00:12:04 --> 00:12:07
interested in,
the exponentials.
217
00:12:07 --> 00:12:12
So -- so that's -- that's the
solution -- what are c1 and c2?
218
00:12:12 --> 00:12:14
Then we're through.
219
00:12:14 --> 00:12:16
What are c1 and c2?
220
00:12:16 --> 00:12:20.46
Well, of course we know these
actual things.
221
00:12:20.46 --> 00:12:23
Let me just -- let me come back
to this.
222
00:12:23 --> 00:12:29
c1 is -- we haven't figured out
yet, but e to the lambda one t,
223
00:12:29 --> 00:12:34
the lambda one is zero so
that's just a one times x1 which
224
00:12:34 --> 00:12:36.47
is two one.
225
00:12:36.47 --> 00:12:42
So it's c1 times this one
that's not moving times the
226
00:12:42 --> 00:12:48
vector, the eigenvector two one
and c2 times -- what's e to the
227
00:12:48 --> 00:12:50
lambda two t?
228
00:12:50 --> 00:12:52.96
Lambda two is minus three.
229
00:12:52.96 --> 00:12:58
So this is the term that has
the minus three t and its
230
00:12:58 --> 00:13:02
eigenvector is this one minus
one.
231
00:13:02 --> 00:13:07.08
So this vector solves the
equation
232
00:13:07.08 --> 00:13:09.04
and any multiple of it.
233
00:13:09.04 --> 00:13:14
This vector solves the equation
if it's got that factor e to the
234
00:13:14 --> 00:13:15
minus three t.
235
00:13:15 --> 00:13:19
We've got the answer except for
c1 and c2.
236
00:13:19 --> 00:13:24
So -- so everything I've done
is immediate as soon as you know
237
00:13:24 --> 00:13:27
the eigenvalues and
eigenvectors.
238
00:13:27 --> 00:13:30
So how do we get c1 and c2?
239
00:13:30 --> 00:13:35
That has to come from the
initial condition.
240
00:13:35 --> 00:13:41
So now I -- now I use -- u of
zero is given as one zero.
241
00:13:41 --> 00:13:48
So this is the initial
condition that will find c1 and
242
00:13:48 --> 00:13:48.56
c2.
243
00:13:48.56 --> 00:13:53
So let me do that on the board
underneath.
244
00:13:53 --> 00:14:00
At t equals zero,
then -- I get c1 times this guy
245
00:14:00 --> 00:14:04
plus c2 and now I'm at times
zero.
246
00:14:04 --> 00:14:11
So that's a one and this is a
one minus one and that's
247
00:14:11 --> 00:14:16
supposed to agree with u of zero
one zero.
248
00:14:16 --> 00:14:17.02
Okay.
249
00:14:17.02 --> 00:14:20
That should be two equations.
250
00:14:20 --> 00:14:26
That should give me c1 and c2
and then I'm through.
251
00:14:26 --> 00:14:30
So what are c1 and
c2?
252
00:14:30 --> 00:14:31
Let's see.
253
00:14:31 --> 00:14:37
I guess we could actually spot
them by eye or we could solve
254
00:14:37 --> 00:14:40
two equations in two unknowns.
255
00:14:40 --> 00:14:41
Let's see.
256
00:14:41 --> 00:14:46
If these were both ones -- so
I'm just adding -- then I would
257
00:14:46 --> 00:14:48
get three zero.
258
00:14:48 --> 00:14:52
So what's the -- what's the
solution, then?
259
00:14:52 --> 00:14:57
If -- if c1 and c2 are both
ones, I get three zero,
260
00:14:57 --> 00:15:00.5
so I want, like,
one third of that,
261
00:15:00.5 --> 00:15:03
because I want to get one zero.
262
00:15:03 --> 00:15:07
So I think it's c1 equals a
third, c2 equals a third.
263
00:15:07 --> 00:15:10.59
So finally I have the answer.
264
00:15:10.59 --> 00:15:14
Let me keep it in the -- in
this board here.
265
00:15:14 --> 00:15:18
Finally the answer is one third
of
266
00:15:18 --> 00:15:21
this plus one third of this.
267
00:15:21 --> 00:15:27
Do you see what -- what's
actually happening with this
268
00:15:27 --> 00:15:28
flow?
269
00:15:28 --> 00:15:34
This flow started out at -- the
solution started out at one
270
00:15:34 --> 00:15:35
zero.
271
00:15:35 --> 00:15:37
Started at one zero.
272
00:15:37 --> 00:15:41
Then as time went on,
people moved,
273
00:15:41 --> 00:15:42
essentially.
274
00:15:42 --> 00:15:47
Some fraction of this
one moved here.
275
00:15:47 --> 00:15:53.1
And -- and in the limit,
there's -- there's the limit,
276
00:15:53.1 --> 00:15:54
as -- right?
277
00:15:54 --> 00:15:58
As t goes to infinity,
as t gets very large,
278
00:15:58 --> 00:16:03
this disappears and this is the
steady state.
279
00:16:03 --> 00:16:09
So the steady state is -- so
the steady state -- u --
280
00:16:09 --> 00:16:15
we could call it u at infinity
is one third of two and one.
281
00:16:15 --> 00:16:19
It's -- it's two thirds of one
third.
282
00:16:19 --> 00:16:24
So that's the -- we really -- I
mean, you're getting,
283
00:16:24 --> 00:16:28
like, total,
insight into the behavior of
284
00:16:28 --> 00:16:32.74
the solution,
what the differential
285
00:16:32.74 --> 00:16:34
equation does.
286
00:16:34 --> 00:16:39
Of course, we don't -- wouldn't
always have a steady state.
287
00:16:39 --> 00:16:42
Sometimes we would approach
zero.
288
00:16:42 --> 00:16:44
Sometimes we would blow up.
289
00:16:44 --> 00:16:48
Can we straighten out those
cases?
290
00:16:48 --> 00:16:50.86
The eigenvalue should tell us.
291
00:16:50.86 --> 00:16:56
So when do we get -- so -- so
let me ask first,
292
00:16:56 --> 00:16:58.6
when do we get stability?
293
00:16:58.6 --> 00:17:01
That's u of t going to zero.
294
00:17:01 --> 00:17:07
When would the solution go to
zero no matter what the initial
295
00:17:07 --> 00:17:08
condition is?
296
00:17:08 --> 00:17:10
Negative eigenvalues,
right.
297
00:17:10 --> 00:17:13
Negative eigenvalues.
298
00:17:13 --> 00:17:19
But now I have to -- I have to
ask you for one more step.
299
00:17:19 --> 00:17:23
Suppose the eigenvalues are
complex numbers?
300
00:17:23 --> 00:17:26
Because we know they could be.
301
00:17:26 --> 00:17:32
Then we want stability -- this
-- this -- we want -- we need
302
00:17:32 --> 00:17:38
all these e to the lambda t-s
all going to zero and somehow
303
00:17:38 --> 00:17:41
that asks us to have lambda
negative.
304
00:17:41 --> 00:17:46
But suppose lambda is a complex
number?
305
00:17:46 --> 00:17:48.23
Then what's the test?
306
00:17:48.23 --> 00:17:52
What -- if lambda's a complex
number like, oh,
307
00:17:52 --> 00:17:56
suppose lambda is negative plus
an imaginary part?
308
00:17:56 --> 00:17:59
Say lambda is minus three plus
six i?
309
00:17:59 --> 00:18:02
What -- what happens then?
310
00:18:02 --> 00:18:06
Can we just,
like, do a -- a case here?
311
00:18:06 --> 00:18:11
If -- if this lambda is minus
three plus six it,
312
00:18:11 --> 00:18:13
how big is that number?
313
00:18:13 --> 00:18:19
Does this -- does this
imaginary part play a -- play a
314
00:18:19 --> 00:18:22
-- play a role here or not?
315
00:18:22 --> 00:18:27
Or how big is -- what's the
absolute value of that -- of
316
00:18:27 --> 00:18:29
that quantity?
317
00:18:29 --> 00:18:34
It's just e to the minus
three t, right?
318
00:18:34 --> 00:18:39
Because this other part,
this -- the -- the magnitude --
319
00:18:39 --> 00:18:44
the -- this -- e to the six it
-- what -- that has absolute
320
00:18:44 --> 00:18:45
value one.
321
00:18:45 --> 00:18:45.74
Right?
322
00:18:45.74 --> 00:18:50
That's just this cosine of six
t plus i, sine of six t.
323
00:18:50 --> 00:18:56
And the absolute value squared
will be cos squared plus sine
324
00:18:56 --> 00:18:58
squared will be one.
325
00:18:58 --> 00:19:03
This is -- this complex number
is running around the unit
326
00:19:03 --> 00:19:03
circle.
327
00:19:03 --> 00:19:08
This com- this -- the -- it's
the real part that matters.
328
00:19:08 --> 00:19:10
This is what I'm trying to do.
329
00:19:10 --> 00:19:13
Real part of lambda has to be
negative.
330
00:19:13 --> 00:19:18
If lambda's a complex number,
it's the real part,
331
00:19:18 --> 00:19:23
the minus three,
that either makes us go to zero
332
00:19:23 --> 00:19:26
or doesn't, or let -- or blows
up.
333
00:19:26 --> 00:19:30
The imaginary part won't --
will just, like,
334
00:19:30 --> 00:19:34
oscillate between the two
components.
335
00:19:34 --> 00:19:34
Okay.
336
00:19:34 --> 00:19:36
So that's stability.
337
00:19:36 --> 00:19:40
And what about -- what about a
steady state?
338
00:19:40 --> 00:19:45
When would we have,
a steady state,
339
00:19:45 --> 00:19:48
always in the same direction?
340
00:19:48 --> 00:19:54
So let me -- I'll take this
part away -- when -- so that
341
00:19:54 --> 00:20:01
was, like, checking that it's --
that it's the real part that we
342
00:20:01 --> 00:20:02
care about.
343
00:20:02 --> 00:20:08
Now, we have a steady state
when -- when lambda one is zero
344
00:20:08 --> 00:20:12
and
the other eigenvalues have
345
00:20:12 --> 00:20:13
what?
346
00:20:13 --> 00:20:18.08
So I'm looking -- like,
that example was,
347
00:20:18.08 --> 00:20:21
like, perfect for a steady
state.
348
00:20:21 --> 00:20:27
We have a zero eigenvalue and
the other eigenvalues,
349
00:20:27 --> 00:20:30
we want those to disappear.
350
00:20:30 --> 00:20:36
So the other eigenvalues have
real part negative.
351
00:20:36 --> 00:20:42
And we blow up,
for sure -- we blow up if any
352
00:20:42 --> 00:20:45
real part of lambda is positive.
353
00:20:45 --> 00:20:51
So if I -- if I reverse the
sign of A -- of the matrix A --
354
00:20:51 --> 00:20:57.13
suppose instead of the matrix I
had, the A that I had,
355
00:20:57.13 --> 00:21:01
I changed it -- I changed all
its sines.
356
00:21:01 --> 00:21:06
What would that do to the
eigenvalues and eigenvectors?
357
00:21:06 --> 00:21:11
If I -- if --
if I know the eigenvalues and
358
00:21:11 --> 00:21:14
eigenvectors of A,
tell me about minus A.
359
00:21:14 --> 00:21:16
What happens to the
eigenvalues?
360
00:21:16 --> 00:21:19
For minus A,
they'll all change sine.
361
00:21:19 --> 00:21:21
So I'll have blow up.
362
00:21:21 --> 00:21:25
This -- instead of the e to the
minus three t,
363
00:21:25 --> 00:21:30
if I change that to minus --
if I change the sines in that
364
00:21:30 --> 00:21:34
matrix, I would change the trace
to plus three,
365
00:21:34 --> 00:21:38
I would have an e to the plus
three t and I would have blow
366
00:21:38 --> 00:21:39.17
up.
367
00:21:39.17 --> 00:21:43
Of course the zero eigenvalue
would stay at zero,
368
00:21:43 --> 00:21:47.71
but the other guy is taking off
in -- if I reversed all the
369
00:21:47.71 --> 00:21:48
sines.
370
00:21:48 --> 00:21:48
Okay.
371
00:21:48 --> 00:21:53
So this is --
this is the stability picture.
372
00:21:53 --> 00:21:59
And for a two by two matrix,
we can actually pin down even
373
00:21:59 --> 00:22:01
more closely what that means.
374
00:22:01 --> 00:22:04
Can I -- let -- can I do that?
375
00:22:04 --> 00:22:09
Let me do that -- I want to --
for a two by two matrix,
376
00:22:09 --> 00:22:15
I can tell whether the real
part of the eigenvalues is
377
00:22:15 --> 00:22:20
negative,
I -- well, let me -- let me
378
00:22:20 --> 00:22:24
tell you what I have in mind for
that.
379
00:22:24 --> 00:22:32
So two by two stability -- let
me -- this is just a little
380
00:22:32 --> 00:22:33
comment.
381
00:22:33 --> 00:22:35.62
Two by two stability.
382
00:22:35.62 --> 00:22:40
So my matrix,
now, is just a b c d and I'm
383
00:22:40 --> 00:22:49
looking for the real parts of
both eigenvalues to be negative.
384
00:22:49 --> 00:22:50
Okay.
385
00:22:50 --> 00:22:55
What -- how can I tell from
looking at the matrix,
386
00:22:55 --> 00:22:59
without computing its
eigenvalues, whether the two
387
00:22:59 --> 00:23:05
eigenvalues are negative,
or at least their real parts
388
00:23:05 --> 00:23:06.35
are negative?
389
00:23:06.35 --> 00:23:10
What would that tell me about
the trace?
390
00:23:10 --> 00:23:15
So -- so the trace --
that's this a plus d -- what
391
00:23:15 --> 00:23:20
can you tell me about the trace
in the case of a two by two
392
00:23:20 --> 00:23:21.96
stable matrix?
393
00:23:21.96 --> 00:23:26
That means the eigenvalues have
-- are negative,
394
00:23:26 --> 00:23:31.09
or at least the real parts of
those eigenvalues are negative
395
00:23:31.09 --> 00:23:34
-- then, when I take the -- when
I
396
00:23:34 --> 00:23:39
look at the matrix and find its
trace, what -- what do I know
397
00:23:39 --> 00:23:40
about that?
398
00:23:40 --> 00:23:42
It's negative,
right.
399
00:23:42 --> 00:23:46
This is the sum of the -- this
equals -- this equals lambda one
400
00:23:46 --> 00:23:49.59
plus lambda two,
so it's negative.
401
00:23:49.59 --> 00:23:53
The two eigenvalues,
by the way, will have --
402
00:23:53 --> 00:23:58.68
if they're complex -- will have
plus six i and minus six i.
403
00:23:58.68 --> 00:24:03
The complex parts will -- will
be conjugates of each other and
404
00:24:03 --> 00:24:07
disappear when we add and the
trace will be negative.
405
00:24:07 --> 00:24:10
Okay, the trace has to be
negative.
406
00:24:10 --> 00:24:15
Is that enough -- is a negative
trace enough to make the matrix
407
00:24:15 --> 00:24:17
stable?
408
00:24:17 --> 00:24:19.08
Shouldn't be enough,
right?
409
00:24:19.08 --> 00:24:23
Can I -- can you make -- what's
a matrix that has a negative
410
00:24:23 --> 00:24:25
trace but still it's not stable?
411
00:24:25 --> 00:24:30
So it -- it has a blow -- it
still has a blow-up factor and a
412
00:24:30 --> 00:24:32
-- and a -- and a decaying one.
413
00:24:32 --> 00:24:37
So what would be a -- so just
-- just to see -- maybe I just
414
00:24:37 --> 00:24:39
put that here.
415
00:24:39 --> 00:24:45
This -- now I'm looking for an
example where the trace could be
416
00:24:45 --> 00:24:48
negative but still blow up.
417
00:24:48 --> 00:24:53
Of course -- yeah,
let's just take one.
418
00:24:53 --> 00:24:59
Oh, look, let me -- let me make
it minus two zero zero one.
419
00:24:59 --> 00:25:00
Okay.
420
00:25:00 --> 00:25:05
There's a case where that
matrix has
421
00:25:05 --> 00:25:09
negative trace -- I know its
eigenvalues of course.
422
00:25:09 --> 00:25:12
They're minus two and one and
it blows up.
423
00:25:12 --> 00:25:15
It's got -- it's got a plus one
eigenvalue here,
424
00:25:15 --> 00:25:20
so there would be an e to the
plus t in the solution and it'll
425
00:25:20 --> 00:25:24
blow up if it has any second
component at all.
426
00:25:24 --> 00:25:26
I need another condition.
427
00:25:26 --> 00:25:29
And it's a condition on the
determinant.
428
00:25:29 --> 00:25:31
And what's that condition?
429
00:25:31 --> 00:25:35.95
If I know that the two
eigenvalues -- suppose I know
430
00:25:35.95 --> 00:25:38
they're negative,
both negative.
431
00:25:38 --> 00:25:41
What does that tell me about
the determinant?
432
00:25:41 --> 00:25:43
Let me ask again.
433
00:25:43 --> 00:25:47
If I know both the eigenvalues
are negative,
434
00:25:47 --> 00:25:52
then I know the trace is
negative but the determinant is
435
00:25:52 --> 00:25:57
positive, because it's the
product of the two eigenvalues.
436
00:25:57 --> 00:26:01
So this determinant is lambda
one times lambda two.
437
00:26:01 --> 00:26:07
This is -- this is lambda one
times lambda two and if they're
438
00:26:07 --> 00:26:11.4
both negative,
the product is positive.
439
00:26:11.4 --> 00:26:15
So positive determinant,
negative trace.
440
00:26:15 --> 00:26:20
I can easily track down the --
this condition also for the --
441
00:26:20 --> 00:26:25
if -- if there's an imaginary
part hanging around.
442
00:26:25 --> 00:26:25
Okay.
443
00:26:25 --> 00:26:29
So that's a -- like a small but
quite useful,
444
00:26:29 --> 00:26:34
because two by two is always
important -- comment on
445
00:26:34 --> 00:26:36
stability.
446
00:26:36 --> 00:26:36
Okay.
447
00:26:36 --> 00:26:40
So let's just look at the
picture again.
448
00:26:40 --> 00:26:40.9
Okay.
449
00:26:40.9 --> 00:26:46
The main part of my lecture,
the one thing you want to be
450
00:26:46 --> 00:26:50
able to, like,
just do automatically is this
451
00:26:50 --> 00:26:52
step of solving the system.
452
00:26:52 --> 00:26:56
Find the eigenvalues,
find the eigenvectors,
453
00:26:56 --> 00:26:58
find the coefficients.
454
00:26:58 --> 00:27:03
And notice --
what's the matrix -- in this
455
00:27:03 --> 00:27:07
linear system here,
I can't help -- if I -- if I
456
00:27:07 --> 00:27:12
have equations like that --
that's my equations column at a
457
00:27:12 --> 00:27:16
time -- what's the matrix form
of that equation?
458
00:27:16 --> 00:27:21
So -- so this -- this system of
equations is --
459
00:27:21 --> 00:27:26
is some matrix multiplying c1,
c2 to give u of zero.
460
00:27:26 --> 00:27:27
One zero.
461
00:27:27 --> 00:27:29
What's the matrix?
462
00:27:29 --> 00:27:33
Well, it's obviously two one,
one minus one,
463
00:27:33 --> 00:27:37
but we have a name,
or at least a letter --
464
00:27:37 --> 00:27:40.57
actually a name for that matrix.
465
00:27:40.57 --> 00:27:46
Wh- what matrix are we s- are
we -- are we dealing with here
466
00:27:46 --> 00:27:50
in this
step of finding the c-s?
467
00:27:50 --> 00:27:55
Its letter is S -- it's the
eigenvector matrix.
468
00:27:55 --> 00:27:56
Of course.
469
00:27:56 --> 00:28:01.85
These are the eigenvectors,
there in the columns of our
470
00:28:01.85 --> 00:28:02
matrix.
471
00:28:02 --> 00:28:08
So this is S c equals u of zero
-- is the -- that step where you
472
00:28:08 --> 00:28:14
find the actual coefficients,
you find out how much of each
473
00:28:14 --> 00:28:18
pure
exponential is in the solution.
474
00:28:18 --> 00:28:23
By getting it right at the
start, then it stays right
475
00:28:23 --> 00:28:23
forever.
476
00:28:23 --> 00:28:27
I -- you're seeing this picture
that each -- each pure
477
00:28:27 --> 00:28:32
exponential goes on its own way
once you start it from u of
478
00:28:32 --> 00:28:32
zero.
479
00:28:32 --> 00:28:37
So you start it by figuring out
how much each one is present in
480
00:28:37 --> 00:28:41
u of
zero and then off they go.
481
00:28:41 --> 00:28:42
Okay.
482
00:28:42 --> 00:28:48
So -- so that's a system of two
equations in two unknowns
483
00:28:48 --> 00:28:54
coupled -- the matrix sort of
couples u1 and u2 and the
484
00:28:54 --> 00:29:01
eigenvalues and eigenvectors
uncouple it, diagonalize it.
485
00:29:01 --> 00:29:07
Actually --
okay, now can I -- can I think
486
00:29:07 --> 00:29:10
in terms of S and lambda?
487
00:29:10 --> 00:29:18
So I want to write the solution
down, again in terms of S and
488
00:29:18 --> 00:29:19
lambda.
489
00:29:19 --> 00:29:20
Okay.
490
00:29:20 --> 00:29:24
I'll do that on this far board.
491
00:29:24 --> 00:29:24
Okay.
492
00:29:24 --> 00:29:31
So I'm coming back to -- I'm
coming back to our equation
493
00:29:31 --> 00:29:33
du/dt equals Au.
494
00:29:33 --> 00:29:39
Now this matrix A couples them.
495
00:29:39 --> 00:29:44
The whole point of eigenvectors
is to uncouple.
496
00:29:44 --> 00:29:50
So one way to see that is
introduce set u equal A -- not
497
00:29:50 --> 00:29:50
A.
498
00:29:50 --> 00:29:55
It's S, the eigenvector matrix
that uncouples it.
499
00:29:55 --> 00:30:00
So I'm -- I'm taking this
equation as I'm given,
500
00:30:00 --> 00:30:06
coupled with -- with A has --
is probably full of
501
00:30:06 --> 00:30:10
non-zeroes, but I'm -- by
uncoupling it,
502
00:30:10 --> 00:30:13
I mean I'm diagonalizing it.
503
00:30:13 --> 00:30:18
If I can get a diagonal matrix,
I'm -- I'm in.
504
00:30:18 --> 00:30:19
Okay.
505
00:30:19 --> 00:30:21
So I plug that in.
506
00:30:21 --> 00:30:23
This is A S v.
507
00:30:23 --> 00:30:25
And this is S dv/dt.
508
00:30:25 --> 00:30:27
S is a constant.
509
00:30:27 --> 00:30:32
It's -- this it the eigenvector
matrix.
510
00:30:32 --> 00:30:36
This is the eigenvector matrix.
511
00:30:36 --> 00:30:37
Okay.
512
00:30:37 --> 00:30:43
Now I'm going to bring S
inverse over here.
513
00:30:43 --> 00:30:45
And what have I got?
514
00:30:45 --> 00:30:53
That combination S inverse A S
is lambda, the diagonal matrix.
515
00:30:53 --> 00:31:02
That's -- that's the point,
that in -- if I'm using the
516
00:31:02 --> 00:31:08
eigenvectors as my basis,
then my system of equations is
517
00:31:08 --> 00:31:09
just diagonal.
518
00:31:09 --> 00:31:15
I -- each -- there's no
coupling anymore -- dv1/dt is
519
00:31:15 --> 00:31:17
lambda one v1.
520
00:31:17 --> 00:31:24
So let's just write that down.
dv1/ dt is lambda one v1 and so
521
00:31:24 --> 00:31:27
on for all n of the equations.
522
00:31:27 --> 00:31:34
It's a system of equations but
they're not connected,
523
00:31:34 --> 00:31:43
so they're easy to solve and
why don't I just write down the
524
00:31:43 --> 00:31:48.29
solution?
v -- well, v is now some e to
525
00:31:48.29 --> 00:31:55
the lambda one t -- well,
okay -- I guess my idea here
526
00:31:55 --> 00:32:01
now is to use,
the natural notation --
527
00:32:01 --> 00:32:07
v of T is e to the lambda tv of
zero.
528
00:32:07 --> 00:32:14
And u of t will be Se to the
lambda t S inverse,
529
00:32:14 --> 00:32:15
u of zero.
530
00:32:15 --> 00:32:23
This is the -- this is the,
formula I'm headed for.
531
00:32:23 --> 00:32:31.07
This -- this matrix,
S e to the lambda t S inverse,
532
00:32:31.07 --> 00:32:35
that's my exponential.
533
00:32:35 --> 00:32:41
That's my e to the A t,
is this S e to the lambda t S
534
00:32:41 --> 00:32:42.21
inverse.
535
00:32:42.21 --> 00:32:48
So my -- my job really now is
to explain what's going on with
536
00:32:48 --> 00:32:51
this matrix up in the
exponential.
537
00:32:51 --> 00:32:54.09
What does that mean?
538
00:32:54.09 --> 00:32:58
What does it mean to say e to a
matrix?
539
00:32:58 --> 00:33:03
This ought to be easier because
this
540
00:33:03 --> 00:33:08
is e to a diagonal matrix,
but still it's a matrix.
541
00:33:08 --> 00:33:11
What do we mean by e to the A
t?
542
00:33:11 --> 00:33:16.85
Because really e to the A t is
my answer here.
543
00:33:16.85 --> 00:33:22
My -- my answer to this
equation is -- this u of t,
544
00:33:22 --> 00:33:27
this is my -- this is my e to
the A t u of zero.
545
00:33:27 --> 00:33:32
So it's -- my job is really now
to
546
00:33:32 --> 00:33:35
say what's -- what does that
mean?
547
00:33:35 --> 00:33:41
What's the exponential of a
matrix and why is that formula
548
00:33:41 --> 00:33:42
correct?
549
00:33:42 --> 00:33:43
Okay.
550
00:33:43 --> 00:33:47
So I'll put that on the board
underneath.
551
00:33:47 --> 00:33:50
What's the exponential of a
matrix?
552
00:33:50 --> 00:33:52
Let me come back here.
553
00:33:52 --> 00:33:56
So I'm talking about matrix
exponentials.
554
00:33:56 --> 00:33:58
e to the At.
555
00:33:58 --> 00:33:58
Okay.
556
00:33:58 --> 00:34:05
How are we going to define
the exponential of a -- of
557
00:34:05 --> 00:34:06
something?
558
00:34:06 --> 00:34:11
The trick -- the idea is -- the
thing to go back to is
559
00:34:11 --> 00:34:17
exponential -- there's a power
series for exponentials.
560
00:34:17 --> 00:34:22
That's how you -- you -- the
good way to define e to
561
00:34:22 --> 00:34:28
the x is the power series one
plus x plus one half x squared,
562
00:34:28 --> 00:34:34.89
one six x cubed and we'll do it
now when the -- when we have a
563
00:34:34.89 --> 00:34:35.59
matrix.
564
00:34:35.59 --> 00:34:39
So the one becomes the
identity, the x is At,
565
00:34:39 --> 00:34:44.7
the x squared is At squared and
I divide by two.
566
00:34:44.7 --> 00:34:48
The cube, the x cube is At
cubed over six,
567
00:34:48 --> 00:34:53
and what's the
general term in here?
568
00:34:53 --> 00:34:59
At to the n-th power divided by
-- and it goes on.
569
00:34:59 --> 00:35:01
But what do I divide by?
570
00:35:01 --> 00:35:06
So, you see the pattern -- here
I divided by one,
571
00:35:06 --> 00:35:13
here I divided by one by two by
six, those are the factorials.
572
00:35:13 --> 00:35:15
It's n factorial.
573
00:35:15 --> 00:35:20
That was, like,
the one beautiful
574
00:35:20 --> 00:35:21.29
Taylor series.
575
00:35:21.29 --> 00:35:26
The one beautiful Taylor series
-- well, there are two -- there
576
00:35:26 --> 00:35:30
are two beautiful Taylor series
in this world.
577
00:35:30 --> 00:35:35
The Taylor series for e to the
x is the s with x to the n-th
578
00:35:35 --> 00:35:36
over n factorial.
579
00:35:36 --> 00:35:42
And all I'm doing is doing the
same thing for matrixes.
580
00:35:42 --> 00:35:46
The other beautiful Taylor
series is just the sum of x to
581
00:35:46 --> 00:35:49
the n-th not divided by n
factorial.
582
00:35:49 --> 00:35:53
Can you -- do you know what
function that one is?
583
00:35:53 --> 00:35:57.73
So if I take -- this is the
series, all these sums are going
584
00:35:57.73 --> 00:35:59
from zero to infinity.
585
00:35:59 --> 00:36:03.13
What's -- what function have I
got --
586
00:36:03.13 --> 00:36:07
this is like a side comment --
this is one plus x plus x
587
00:36:07 --> 00:36:12
squared plus x cubed plus x to
the fourth not divided by
588
00:36:12 --> 00:36:15
anything, what's -- what's that
function?
589
00:36:15 --> 00:36:20.54
One plus x plus x squared plus
x cubed plus x fourth forever is
590
00:36:20.54 --> 00:36:22
one over one minus x.
591
00:36:22 --> 00:36:27
It's the geometric series,
the nicest power series of all.
592
00:36:27 --> 00:36:31
So, actually,
of course, there would be a
593
00:36:31 --> 00:36:33
similar thing here.
594
00:36:33 --> 00:36:37
If -- if I wanted,
I minus A t inverse would be --
595
00:36:37 --> 00:36:39
now I've got matrixes.
596
00:36:39 --> 00:36:44
I've got matrixes everywhere,
but it's just like that series
597
00:36:44 --> 00:36:50
with -- and just like this one
without the divisions.
598
00:36:50 --> 00:36:56
It's I plus At plus At squared
plus At cubed and forever.
599
00:36:56 --> 00:37:02
So that's actually a -- a
reasonable way to find the
600
00:37:02 --> 00:37:04
inverse of a matrix.
601
00:37:04 --> 00:37:10
If I chop it off -- well,
it's reasonable if t is small.
602
00:37:10 --> 00:37:15
If t is a small number,
then -- then t squared is
603
00:37:15 --> 00:37:21
extremely small,
t cubed is even smaller,
604
00:37:21 --> 00:37:24.6
so approximately that inverse
is I plus At.
605
00:37:24.6 --> 00:37:26
I can keep more terms if I
like.
606
00:37:26 --> 00:37:28
Do you see what I'm doing?
607
00:37:28 --> 00:37:33
I'm just saying we can do the
same thing for matrixes that we
608
00:37:33 --> 00:37:37
do for ordinary functions and
the good thing about the
609
00:37:37 --> 00:37:41
exponential series -- so in a
way, this series is better than
610
00:37:41 --> 00:37:42
this one.
611
00:37:42 --> 00:37:43
Why?
612
00:37:43 --> 00:37:46
Because this one always
converges.
613
00:37:46 --> 00:37:51
I'm dividing by these bigger
and bigger numbers,
614
00:37:51 --> 00:37:55
so whatever matrix A and
however large t is,
615
00:37:55 --> 00:37:58
that series -- these terms go
to zero.
616
00:37:58 --> 00:38:03
The series adds up to a finite
sum,
617
00:38:03 --> 00:38:07
e to the At is a -- is -- is
completely defined.
618
00:38:07 --> 00:38:10.92
Whereas this second guy could
fail, right?
619
00:38:10.92 --> 00:38:16
If At is big -- somehow if At
has an eigenvalue larger than
620
00:38:16 --> 00:38:21.52
one, then when I square it it'll
have that eigenvalue squared,
621
00:38:21.52 --> 00:38:25
when I cube it the
eigenvalue will be cubed --
622
00:38:25 --> 00:38:30
that series will blow up unless
the eigenvalues of At are
623
00:38:30 --> 00:38:31
smaller than one.
624
00:38:31 --> 00:38:35
So when the eigenvalues of At
are smaller than one -- so I'd
625
00:38:35 --> 00:38:37.25
better put that in.
626
00:38:37.25 --> 00:38:41
The -- all eigenvalues of At
below one -- then that series
627
00:38:41 --> 00:38:45
converges and gives
me the inverse.
628
00:38:45 --> 00:38:45
Okay.
629
00:38:45 --> 00:38:50
So this is the guy I'm chiefly
interested in,
630
00:38:50 --> 00:38:54
and I wanted to connect it to
-- oh, okay.
631
00:38:54 --> 00:38:59.92
What's -- how do I -- how do I
get -- this is my,
632
00:38:59.92 --> 00:39:06
like, main thing now to do --
how do I get e to the At --
633
00:39:06 --> 00:39:11.58
how do I see that e to the At
is the same as this?
634
00:39:11.58 --> 00:39:15
In other words,
I can find e to the At by
635
00:39:15 --> 00:39:20
finding S and lambda,
because now e to the lambda t
636
00:39:20 --> 00:39:20
is easy.
637
00:39:20 --> 00:39:26.41
Lambda's a diagonal matrix and
we can write down either the
638
00:39:26.41 --> 00:39:30
lambda t -- and will right -- in
a minute.
639
00:39:30 --> 00:39:34
But how --
do you see what -- do you see
640
00:39:34 --> 00:39:39
that we're hoping for a -- we're
hoping that we can compute
641
00:39:39 --> 00:39:44
either the A T from S and lambda
-- and I have to look at that
642
00:39:44 --> 00:39:48
definition and say,
okay, how do -- how do I get an
643
00:39:48 --> 00:39:52
S and the lambda to come out of
that?
644
00:39:52 --> 00:39:58
Okay, can -- do you see how I
-- I want to connect that to
645
00:39:58 --> 00:40:01
that, from that definition.
646
00:40:01 --> 00:40:06
So let me erase this -- the
geometric series,
647
00:40:06 --> 00:40:13
which isn't part of the
differential equations case and
648
00:40:13 --> 00:40:18
get the S and the lambda into
this picture.
649
00:40:18 --> 00:40:19
Oh, okay.
650
00:40:19 --> 00:40:21
Here we go.
651
00:40:21 --> 00:40:23
So identity is fine.
652
00:40:23 --> 00:40:28
Now -- all right,
you -- you -- you'll see how
653
00:40:28 --> 00:40:35
I'm -- how I'm -- how I going to
get A replaced by S and S is in
654
00:40:35 --> 00:40:36
lambda's?
655
00:40:36 --> 00:40:42
Well I use the fundamental
formula of this whole chapter.
656
00:40:42 --> 00:40:46
A is S lambda S inverse and
then times t.
657
00:40:46 --> 00:40:47
That's At.
658
00:40:47 --> 00:40:49
Okay.
659
00:40:49 --> 00:40:51
What's A squared t?
660
00:40:51 --> 00:40:56
I can -- I've got the divide by
two, I've got the t squared and
661
00:40:56 --> 00:40:59
I've got an A squared.
662
00:40:59 --> 00:41:04
All right, I -- so I've got a
-- there's A -- there's A.
663
00:41:04 --> 00:41:05
Now square it.
664
00:41:05 --> 00:41:08
So what happens when I square
it?
665
00:41:08 --> 00:41:11
We've seen that before.
666
00:41:11 --> 00:41:15
When I square it,
I get S lambda squared S
667
00:41:15 --> 00:41:16
inverse, right?
668
00:41:16 --> 00:41:21
When I square that thing,
the -- there's an S and a -- an
669
00:41:21 --> 00:41:24.34
S cancels out an S inverse.
670
00:41:24.34 --> 00:41:29
I'm left with the S on the
left, the S inverse on the right
671
00:41:29 --> 00:41:32
and lambda squared in the
middle.
672
00:41:32 --> 00:41:34
And so on.
673
00:41:34 --> 00:41:40
The next one'll be S lambda
cubed, S inverse -- times t
674
00:41:40 --> 00:41:42
cubed over three factorial.
675
00:41:42 --> 00:41:45.63
And now -- what do I do now?
676
00:41:45.63 --> 00:41:49
I want to pull an S out from
everything.
677
00:41:49 --> 00:41:53
I want an S out of the whole
thing.
678
00:41:53 --> 00:41:57
Well, look, I'd better write
the identity how?
679
00:41:57 --> 00:42:02
I -- I want to be able to pull
an S
680
00:42:02 --> 00:42:05.99
out and an S inverse out from
the other side,
681
00:42:05.99 --> 00:42:10
so I just write the identity as
S times S inverse.
682
00:42:10 --> 00:42:15
So I have an S there and an S
inverse from this side and what
683
00:42:15 --> 00:42:17
have I got in the middle?
684
00:42:17 --> 00:42:22
If I pull out an S and an S
inverse, what have I got in the
685
00:42:22 --> 00:42:23
middle?
686
00:42:23 --> 00:42:28
I've got the identity,
a lambda t, a lambda squared t
687
00:42:28 --> 00:42:33
squared over two -- I've got e
to the lambda t.
688
00:42:33 --> 00:42:36
That's what's in the middle.
689
00:42:36 --> 00:42:39
That's my formula for e to the
At.
690
00:42:39 --> 00:42:41
Oh, now I have to ask you.
691
00:42:41 --> 00:42:44.91
Does this formula always work?
692
00:42:44.91 --> 00:42:50
This formula always works --
well, except it's an infinite
693
00:42:50 --> 00:42:51
series.
694
00:42:51 --> 00:42:55
But what do I mean by always
work?
695
00:42:55 --> 00:43:00
And this one doesn't always
work and I just have to remind
696
00:43:00 --> 00:43:05
you of what assumption is built
into this formula that's not
697
00:43:05 --> 00:43:07
built into the original.
698
00:43:07 --> 00:43:10
The assumption that A can be
diagonalized.
699
00:43:10 --> 00:43:16
You'll remember that there are
some small -- sm- some subset of
700
00:43:16 --> 00:43:21
matrixes that don't have n
independent eigenvectors,
701
00:43:21 --> 00:43:25
so we don't have an S inverse
for those matrixes and the whole
702
00:43:25 --> 00:43:27
diagonalization breaks down.
703
00:43:27 --> 00:43:30
We could still make it
triangular.
704
00:43:30 --> 00:43:31
I'll tell you about that.
705
00:43:31 --> 00:43:36
But diagonal we can't do for
those particular degenerate
706
00:43:36 --> 00:43:40
matrixes that don't have enough
independent eigenvectors.
707
00:43:40 --> 00:43:43
But otherwise,
this is golden.
708
00:43:43 --> 00:43:44.13
Okay.
709
00:43:44.13 --> 00:43:49
So that's the formula -- that's
the matrix exponential.
710
00:43:49 --> 00:43:54
Now it just remains for me to
say what is e to the lambda t?
711
00:43:54 --> 00:43:56
Can I just do that?
712
00:43:56 --> 00:43:59
Let me just put that in the
corner here.
713
00:43:59 --> 00:44:05
What is the exponential of a
diagonal matrix?
714
00:44:05 --> 00:44:10
Remember lambda is diagonal,
lambda one down to lambda n.
715
00:44:10 --> 00:44:15
What's the exponential of that
diagonal matrix?
716
00:44:15 --> 00:44:20
Because our whole point is that
this ought to be simple.
717
00:44:20 --> 00:44:26.39
Our whole point is that to take
the exponential of a diagonal
718
00:44:26.39 --> 00:44:31
matrix ought to be completely
decoupled --
719
00:44:31 --> 00:44:34
it ought to be diagonal,
in other words,
720
00:44:34 --> 00:44:35.88
and it is.
721
00:44:35.88 --> 00:44:40
It's just e to the lambda one
t, e to the lambda two t,
722
00:44:40 --> 00:44:43
e to the lambda n t,
all zeroes.
723
00:44:43 --> 00:44:49
So -- so if we have a diagonal
matrix and I plug it into this
724
00:44:49 --> 00:44:53
exponential formula,
everything's diagonal and the
725
00:44:53 --> 00:44:58
diagonal terms are just the
ordinary
726
00:44:58 --> 00:45:01.89
scaler exponentials e to the
lambda one t.
727
00:45:01.89 --> 00:45:07
Okay, so that's -- that's -- in
a sense, I'm doing here,
728
00:45:07 --> 00:45:09
on this board,
with -- with,
729
00:45:09 --> 00:45:15
like, formulas what I did on
the -- in the first half of the
730
00:45:15 --> 00:45:20
lecture with specific matrix A
and specific
731
00:45:20 --> 00:45:22
eigenvalues and eigenvectors.
732
00:45:22 --> 00:45:26
The -- so let me show you the
formulas again.
733
00:45:26 --> 00:45:31
But the -- so you -- I guess --
what should you know from this
734
00:45:31 --> 00:45:31
lecture?
735
00:45:31 --> 00:45:36
You should know what this
matrix exponential is and,
736
00:45:36 --> 00:45:38
like, when does it go to zero?
737
00:45:38 --> 00:45:41
Tell me again,
now,
738
00:45:41 --> 00:45:42
the answer to that.
739
00:45:42 --> 00:45:47
When does e to the At approach
-- get smaller and smaller as t
740
00:45:47 --> 00:45:48
increases?
741
00:45:48 --> 00:45:51
Well, the S and the S inverse
aren't moving.
742
00:45:51 --> 00:45:56
It's this one that has to get
smaller and smaller and that one
743
00:45:56 --> 00:45:59
has this simple diagonal form.
744
00:45:59 --> 00:46:02.96
And it goes to zero provided
every
745
00:46:02.96 --> 00:46:08
one of these lambdas -- I -- I
need to have each one of these
746
00:46:08 --> 00:46:11
guys go to zero,
so I need every real part of
747
00:46:11 --> 00:46:13
every eigenvalue negative.
748
00:46:13 --> 00:46:14
Right?
749
00:46:14 --> 00:46:18
If the real part is negative,
that's -- that takes the
750
00:46:18 --> 00:46:23
exponential -- that forces --
the exponential goes to zero.
751
00:46:23 --> 00:46:28
Okay, so that -- that's
really the difference.
752
00:46:28 --> 00:46:34
If I can just draw the --
here's a picture of the -- of
753
00:46:34 --> 00:46:37
the -- this is the complex
plain.
754
00:46:37 --> 00:46:42.22
Here's the real axis and here's
the imaginary axis.
755
00:46:42.22 --> 00:46:47.65
And where do the eigenvalues
have to be for stability in
756
00:46:47.65 --> 00:46:49
differential equations?
757
00:46:49 --> 00:46:55
They have to be over here,
in the left half plain.
758
00:46:55 --> 00:47:00
So the left half plain is this
plain, real part of lambda,
759
00:47:00 --> 00:47:01.48
less than zero.
760
00:47:01.48 --> 00:47:06
Where do the ma- where do the
eigenvalues have to be for
761
00:47:06 --> 00:47:08
powers of the matrix to go to
zero?
762
00:47:08 --> 00:47:14
Powers of the matrix go to zero
if the eigenvalues are in here.
763
00:47:14 --> 00:47:19.29
So this is the stability region
for powers -- this is the region
764
00:47:19.29 --> 00:47:22
--
absolute value of lambda,
765
00:47:22 --> 00:47:23
less than one.
766
00:47:23 --> 00:47:29
That's the stability for --
that tells us that the powers of
767
00:47:29 --> 00:47:32
A go to zero,
this tells us that the
768
00:47:32 --> 00:47:34
exponential of A goes to zero.
769
00:47:34 --> 00:47:35
Okay.
770
00:47:35 --> 00:47:36
One final example.
771
00:47:36 --> 00:47:42
Let me just write down how to
deal with a final example.
772
00:47:42 --> 00:47:44
Let's see.
773
00:47:44 --> 00:47:50
So my final example will be a
single equation,
774
00:47:50 --> 00:47:52
u''+bu'+Ku=0.
775
00:47:52 --> 00:47:56
One equation,
second order.
776
00:47:56 --> 00:48:04
How do I -- and maybe I should
have used -- I'll use -- I
777
00:48:04 --> 00:48:13
prefer to use y here,
because that's what you see in
778
00:48:13 --> 00:48:16
differential equations.
779
00:48:16 --> 00:48:20.13
And I want u to be a vector.
780
00:48:20.13 --> 00:48:27
So how do I change one second
order equation into a two by two
781
00:48:27 --> 00:48:30
first order system?
782
00:48:30 --> 00:48:34
Just the way I did for
Fibonacci.
783
00:48:34 --> 00:48:38
I'll let u be y prime and y.
784
00:48:38 --> 00:48:45
What I'm going to do is I'm
going to add an extra equation,
785
00:48:45 --> 00:48:49
y prime equals y prime.
786
00:48:49 --> 00:48:55
So I take this -- so by -- so
using this as the vector
787
00:48:55 --> 00:48:58.67
unknown, now my equation is u
prime.
788
00:48:58.67 --> 00:49:04
My first order system is u
prime, that'll be y double prime
789
00:49:04 --> 00:49:09
y prime, the derivative of u,
okay, now the differential
790
00:49:09 --> 00:49:15
equation is telling me that y
double prime is m- so I'm just
791
00:49:15 --> 00:49:20
looking for -- what's this
matrix?
792
00:49:20 --> 00:49:21.39
y prime y.
793
00:49:21.39 --> 00:49:24
I'm looking for the matrix A.
794
00:49:24 --> 00:49:29.4
What's the matrix in case I
have a single first order
795
00:49:29.4 --> 00:49:34
equation and I want to make it
into a two by two system?
796
00:49:34 --> 00:49:36
Okay, simple.
797
00:49:36 --> 00:49:41
The first row of the matrix is
given by the equation.
798
00:49:41 --> 00:49:44
So y''-by'-ky
-- no problem.
799
00:49:44 --> 00:49:48
And what's the second row on
the matrix?
800
00:49:48 --> 00:49:50
Then we're done.
801
00:49:50 --> 00:49:55
The second row of the matrix I
want just to be the trivial
802
00:49:55 --> 00:50:01.48
equation y prime equals y prime,
so I need a one and a zero
803
00:50:01.48 --> 00:50:02
there.
804
00:50:02 --> 00:50:06
So matrixes like these,
the gen- the general case,
805
00:50:06 --> 00:50:13
if I had a five by five --
if I had a fifth order equation
806
00:50:13 --> 00:50:18
and I wanted a five by five
matrix, I would see the
807
00:50:18 --> 00:50:24.78
coefficients of the equation up
there and then my four trivial
808
00:50:24.78 --> 00:50:27
equations would put ones here.
809
00:50:27 --> 00:50:33.7
This is the kind of matrix that
goes from a fifth order to a
810
00:50:33.7 --> 00:50:36
five by five first order.
811
00:50:36 --> 00:50:40
So the -- and the eigenvalues
will
812
00:50:40 --> 00:50:46
come out in a natural way
connected to the differential
813
00:50:46 --> 00:50:46
equation.
814
00:50:46 --> 00:50:50.49
Okay, that's differential
equations.
815
00:50:50.49 --> 00:50:56
The -- a parallel lecture
compared to powers of a matrix
816
00:50:56 --> 00:50:58
we can now do exponentials.
817
00:50:58 --> 00:51:01
Thanks.