1
00:00:00,000 --> 00:00:11,590
-- and lift-off on
differential equations.
2
00:00:11,590 --> 00:00:16,660
So, this section is
about how to solve
3
00:00:16,660 --> 00:00:22,760
a system of first order,
first derivative, constant
4
00:00:22,760 --> 00:00:26,500
coefficient linear equations.
5
00:00:26,500 --> 00:00:32,740
And if we do it right, it turns
directly into linear algebra.
6
00:00:32,740 --> 00:00:37,690
The key idea is the solutions
to constant coefficient
7
00:00:37,690 --> 00:00:41,730
linear equations
are exponentials.
8
00:00:41,730 --> 00:00:44,520
So if you look for
an exponential,
9
00:00:44,520 --> 00:00:48,230
then all you have to find is
what's up there in the exponent
10
00:00:48,230 --> 00:00:51,060
and what multiplies
the exponential
11
00:00:51,060 --> 00:00:53,130
and that's the linear algebra.
12
00:00:53,130 --> 00:00:56,950
So -- and the result --
one thing we will fine --
13
00:00:56,950 --> 00:01:01,380
it's completely parallel
to powers of a matrix.
14
00:01:01,380 --> 00:01:06,090
So the last lecture was
about how would you compute
15
00:01:06,090 --> 00:01:08,680
A to the K or A to the 100?
16
00:01:08,680 --> 00:01:12,120
How do you compute high
powers of a matrix?
17
00:01:12,120 --> 00:01:18,590
Now it's not powers anymore,
but it's exponentials.
18
00:01:18,590 --> 00:01:22,310
That's the natural thing
for differential equation.
19
00:01:22,310 --> 00:01:22,810
Okay.
20
00:01:22,810 --> 00:01:26,730
But can I begin with an example?
21
00:01:26,730 --> 00:01:28,630
And I'll just go
through the mechanics.
22
00:01:28,630 --> 00:01:31,700
How would I solve
the differential --
23
00:01:31,700 --> 00:01:33,707
two differential equations?
24
00:01:33,707 --> 00:01:34,790
So I'm going to make it --
25
00:01:34,790 --> 00:01:38,160
I'll have a two by two
matrix and the coefficients
26
00:01:38,160 --> 00:01:42,530
are minus one two, one minus
two and I'd better give you
27
00:01:42,530 --> 00:01:44,200
some initial condition.
28
00:01:44,200 --> 00:01:50,570
So suppose it starts u at
times zero -- this is u1, u2 --
29
00:01:50,570 --> 00:01:52,810
let it -- let it --
30
00:01:52,810 --> 00:01:57,150
suppose everything is
in u1 at times zero.
31
00:01:57,150 --> 00:02:00,890
So -- at -- at the
start, it's all in u1.
32
00:02:00,890 --> 00:02:07,640
But what happens as time
goes on, du2/dt will --
33
00:02:07,640 --> 00:02:10,690
will be positive,
because of that u1 term,
34
00:02:10,690 --> 00:02:16,210
so flow will move into the u2
component and it will go out
35
00:02:16,210 --> 00:02:19,050
of the u1 component.
36
00:02:19,050 --> 00:02:22,330
So we'll just follow
that movement as time
37
00:02:22,330 --> 00:02:28,260
goes forward by looking at the
eigenvalues and eigenvectors
38
00:02:28,260 --> 00:02:30,220
of that matrix.
39
00:02:30,220 --> 00:02:31,430
That's a first job.
40
00:02:31,430 --> 00:02:34,210
Before you do anything
else, find the --
41
00:02:34,210 --> 00:02:39,510
find the matrix and its
eigenvalues and eigenvectors.
42
00:02:39,510 --> 00:02:42,400
So let me do that.
43
00:02:42,400 --> 00:02:43,060
Okay.
44
00:02:43,060 --> 00:02:45,440
So here's our matrix.
45
00:02:45,440 --> 00:02:47,500
Maybe you can tell
me right away what --
46
00:02:47,500 --> 00:02:53,190
what are the eigenvalues
and -- eigenvalues anyway.
47
00:02:53,190 --> 00:02:54,640
And then we can check.
48
00:02:54,640 --> 00:02:58,070
But can you spot any of the
eigenvalues of that matrix?
49
00:02:58,070 --> 00:03:00,900
We're looking for
two eigenvalues.
50
00:03:00,900 --> 00:03:02,090
Do you see --
51
00:03:02,090 --> 00:03:04,530
I mean, if I just wrote
that matrix down, what --
52
00:03:04,530 --> 00:03:07,670
what do you notice about it?
53
00:03:07,670 --> 00:03:09,490
It's singular, right.
54
00:03:09,490 --> 00:03:11,380
That -- that's a
singular matrix.
55
00:03:11,380 --> 00:03:14,190
That tells me right away
that one of the eigenvalues
56
00:03:14,190 --> 00:03:18,860
is lambda equals zero.
57
00:03:18,860 --> 00:03:20,900
I can -- that's a
singular matrix,
58
00:03:20,900 --> 00:03:25,440
the second column is minus
two times the first column,
59
00:03:25,440 --> 00:03:28,600
the determinant is zero,
it's -- it's singular,
60
00:03:28,600 --> 00:03:33,670
so zero is an eigenvalue and
the other eigenvalue will be --
61
00:03:33,670 --> 00:03:35,640
from the trace.
62
00:03:35,640 --> 00:03:37,510
I look at the
trace, the sum down
63
00:03:37,510 --> 00:03:39,620
the diagonal is minus three.
64
00:03:39,620 --> 00:03:42,770
That has to agree with
the sum of the eigenvalue,
65
00:03:42,770 --> 00:03:46,970
so that second eigenvalue
better be minus three.
66
00:03:46,970 --> 00:03:48,610
I could, of course --
67
00:03:48,610 --> 00:03:51,100
I could compute -- why
don't I over here --
68
00:03:51,100 --> 00:03:54,530
compute the determinant
of A minus lambda I,
69
00:03:54,530 --> 00:04:01,220
the determinant of this minus
one minus lambda two one minus
70
00:04:01,220 --> 00:04:03,680
two minus lambda matrix.
71
00:04:03,680 --> 00:04:05,930
But we know what's coming.
72
00:04:05,930 --> 00:04:09,700
When I do that multiplication,
I get a lambda squared.
73
00:04:09,700 --> 00:04:14,200
I get a two lambda and a one
lambda, that's a three lambda.
74
00:04:14,200 --> 00:04:16,750
And then -- now I'm going
to get the determinant,
75
00:04:16,750 --> 00:04:19,910
which is two minus
two which is zero.
76
00:04:19,910 --> 00:04:26,490
So there's my characteristic
polynomial, this determinant.
77
00:04:26,490 --> 00:04:31,960
And of course I factor that into
lambda times lambda plus three
78
00:04:31,960 --> 00:04:37,590
and I get the two eigenvalues
that we saw coming.
79
00:04:37,590 --> 00:04:38,830
What else do I need?
80
00:04:38,830 --> 00:04:40,320
The eigenvectors.
81
00:04:40,320 --> 00:04:44,020
So before I even think about the
differential equation or what
82
00:04:44,020 --> 00:04:47,780
-- how to solve it, let me
find the eigenvectors for this
83
00:04:47,780 --> 00:04:48,910
matrix.
84
00:04:48,910 --> 00:04:49,430
Okay.
85
00:04:49,430 --> 00:04:52,970
So take lambda equals zero --
86
00:04:52,970 --> 00:04:55,370
so that -- that's
the first eigenvalue.
87
00:04:55,370 --> 00:04:58,690
Lambda one equals zero
and the second eigenvalue
88
00:04:58,690 --> 00:05:03,130
will be lambda two
equals minus three.
89
00:05:03,130 --> 00:05:05,210
By the way, I --
90
00:05:05,210 --> 00:05:09,010
I already know something
important about this.
91
00:05:09,010 --> 00:05:11,920
The eigenvalues are
telling me something.
92
00:05:11,920 --> 00:05:15,820
You'll see how it comes
out, but let me point to --
93
00:05:15,820 --> 00:05:19,640
these numbers are
-- this eigenvalue,
94
00:05:19,640 --> 00:05:23,700
a negative eigenvalue,
is going to disappear.
95
00:05:23,700 --> 00:05:28,620
There's going to be an e to the
minus three t in the answer.
96
00:05:28,620 --> 00:05:31,690
That e to the minus
three t as times goes on
97
00:05:31,690 --> 00:05:34,070
is going to be very, very small.
98
00:05:34,070 --> 00:05:38,360
The other part of the
answer will involve an e
99
00:05:38,360 --> 00:05:40,390
to the zero t.
100
00:05:40,390 --> 00:05:44,330
But e to the zero t is
one and that's a constant.
101
00:05:44,330 --> 00:05:49,190
So I'm expecting that this
solution'll have two parts,
102
00:05:49,190 --> 00:05:53,570
an e to the zero t part and an
e to the minus three t part,
103
00:05:53,570 --> 00:05:57,900
and that -- and as time goes
on, the second part'll disappear
104
00:05:57,900 --> 00:06:00,360
and the first part
will be a steady
105
00:06:00,360 --> 00:06:02,030
It won't move. state.
106
00:06:02,030 --> 00:06:06,740
It will be -- at the end of
-- as t approaches infinity,
107
00:06:06,740 --> 00:06:09,800
this part disappears
and this is the --
108
00:06:09,800 --> 00:06:13,330
the e to the zero t
part is what I get.
109
00:06:13,330 --> 00:06:17,120
And I'm very interested in these
steady states, so that's --
110
00:06:17,120 --> 00:06:19,330
I get a steady state
when I have a zero
111
00:06:19,330 --> 00:06:20,150
eigenvalue.
112
00:06:20,150 --> 00:06:20,650
Okay.
113
00:06:20,650 --> 00:06:22,950
What about those eigenvectors?
114
00:06:22,950 --> 00:06:26,170
So what's the eigenvector that
goes with eigenvalue zero?
115
00:06:26,170 --> 00:06:26,670
Okay.
116
00:06:26,670 --> 00:06:31,280
The matrix is singular as
it is, the eigenvector is --
117
00:06:31,280 --> 00:06:34,940
is the guy in the null space,
so what vector is in the null
118
00:06:34,940 --> 00:06:37,890
space of that matrix?
119
00:06:37,890 --> 00:06:38,860
Let's see.
120
00:06:38,860 --> 00:06:42,900
I guess I probably give the
free variable the value one
121
00:06:42,900 --> 00:06:48,630
and I realize that if I want to
get zero I need a two up here.
122
00:06:48,630 --> 00:06:49,260
Okay?
123
00:06:49,260 --> 00:06:52,560
So Ax1 is zero x1.
124
00:06:52,560 --> 00:06:55,550
A x1 is zero x1.
125
00:06:55,550 --> 00:06:56,370
Fine.
126
00:06:56,370 --> 00:06:57,520
Okay.
127
00:06:57,520 --> 00:06:59,520
What about the other eigenvalue?
128
00:06:59,520 --> 00:07:01,801
Lambda two is minus three.
129
00:07:01,801 --> 00:07:02,300
Okay.
130
00:07:02,300 --> 00:07:05,010
How do I get the other
eigenvalue, then?
131
00:07:05,010 --> 00:07:08,080
For the moment --
can I mentally do it?
132
00:07:08,080 --> 00:07:12,850
I subtract minus three
along the diagonal,
133
00:07:12,850 --> 00:07:15,070
which means I add three --
134
00:07:15,070 --> 00:07:18,370
can I -- I'll just do it with an
erase -- erase for the moment.
135
00:07:18,370 --> 00:07:20,950
So I'm going to add
three to the diagonal.
136
00:07:20,950 --> 00:07:26,040
So this minus one will
become a two and --
137
00:07:26,040 --> 00:07:28,450
I'll make it in big
loopy letters --
138
00:07:28,450 --> 00:07:32,350
and when I add three to this
guy, the minus two becomes --
139
00:07:32,350 --> 00:07:36,200
well, I can't make one
very loopy, but how's that?
140
00:07:36,200 --> 00:07:37,410
Okay.
141
00:07:37,410 --> 00:07:40,200
Now that's A minus three I --
142
00:07:40,200 --> 00:07:41,540
A plus three I, sorry.
143
00:07:41,540 --> 00:07:43,400
That's A plus three I.
144
00:07:43,400 --> 00:07:45,510
It's supposed to
be singular, right?
145
00:07:45,510 --> 00:07:48,260
I-- if things --
if I did it right,
146
00:07:48,260 --> 00:07:51,930
this matrix should be
singular and the x2,
147
00:07:51,930 --> 00:07:55,521
the eigenvector should
be in its null space.
148
00:07:55,521 --> 00:07:56,020
Okay.
149
00:07:56,020 --> 00:07:58,510
What do I get for the
null space of this?
150
00:07:58,510 --> 00:08:02,140
Maybe minus one one,
or one minus one.
151
00:08:02,140 --> 00:08:03,240
Doesn't matter.
152
00:08:03,240 --> 00:08:05,300
Those are both perfectly good.
153
00:08:05,300 --> 00:08:05,800
Right?
154
00:08:05,800 --> 00:08:07,550
Because that's in the
null space of this.
155
00:08:07,550 --> 00:08:14,370
Now I'll -- because A times
that vector is three times that
156
00:08:14,370 --> 00:08:15,600
vector.
157
00:08:15,600 --> 00:08:20,370
Ax2 is minus three x2.
158
00:08:20,370 --> 00:08:22,540
Good.
159
00:08:22,540 --> 00:08:23,040
Okay.
160
00:08:23,040 --> 00:08:26,380
Can I get A again so
we see that correctly?
161
00:08:26,380 --> 00:08:29,680
That was a minus one and
that was a minus two.
162
00:08:29,680 --> 00:08:31,730
Good.
163
00:08:31,730 --> 00:08:32,809
Okay.
164
00:08:32,809 --> 00:08:35,470
That -- that's the first job.
165
00:08:35,470 --> 00:08:37,970
eigenvalues and eigenvectors.
166
00:08:37,970 --> 00:08:41,179
And already the
eigenvalues are telling me
167
00:08:41,179 --> 00:08:44,240
the most important
information about the answer.
168
00:08:44,240 --> 00:08:46,540
But now, what is the answer?
169
00:08:46,540 --> 00:08:54,420
The answer is -- the
solution will be U of T --
170
00:08:54,420 --> 00:08:54,920
okay.
171
00:08:54,920 --> 00:08:59,750
Now, wh- now I use those
eigenvalues and eigenvectors.
172
00:08:59,750 --> 00:09:04,210
The solution is some --
there are two eigenvalues.
173
00:09:04,210 --> 00:09:08,100
So I -- it -- so there're going
to be two special solutions
174
00:09:08,100 --> 00:09:08,910
here.
175
00:09:08,910 --> 00:09:12,040
Two pure exponential solutions.
176
00:09:12,040 --> 00:09:17,300
The first one is going to
be either the lambda one tx1
177
00:09:17,300 --> 00:09:23,440
and the -- so that solves the
equation, and so does this one.
178
00:09:23,440 --> 00:09:29,290
They both are solutions to
the differential equation.
179
00:09:29,290 --> 00:09:31,640
That's the general solution.
180
00:09:31,640 --> 00:09:33,900
The general solution
is a combination
181
00:09:33,900 --> 00:09:37,010
of that pure
exponential solution
182
00:09:37,010 --> 00:09:39,630
and that pure
exponential solution.
183
00:09:39,630 --> 00:09:43,150
Can I just see that those
guys do solve the equation?
184
00:09:43,150 --> 00:09:47,540
So let me just check -- check
on this one, for example.
185
00:09:47,540 --> 00:09:53,360
I -- I want to check that
the -- my equation --
186
00:09:53,360 --> 00:09:53,860
let's
187
00:09:53,860 --> 00:09:57,190
Check. remember, the
equation -- du/dt is Au.
188
00:09:57,190 --> 00:10:04,340
I plug in e to the
lambda one t x1
189
00:10:04,340 --> 00:10:08,470
and let's just see that
the equation's okay.
190
00:10:08,470 --> 00:10:12,370
I believe this is a
solution to that equation.
191
00:10:12,370 --> 00:10:13,980
So just plug it in.
192
00:10:13,980 --> 00:10:17,870
On the left-hand side, I
take the time derivative --
193
00:10:17,870 --> 00:10:22,700
so the left-hand side will be
lambda one, e to the lambda one
194
00:10:22,700 --> 00:10:25,540
t x1, right?
195
00:10:25,540 --> 00:10:28,990
The time derivative -- this is
the term that depends on time,
196
00:10:28,990 --> 00:10:33,250
it's just ordinary exponential,
its derivative brings down
197
00:10:33,250 --> 00:10:34,470
a lambda one.
198
00:10:34,470 --> 00:10:37,460
On the other side of the
equation it's A times this
199
00:10:37,460 --> 00:10:38,380
thing.
200
00:10:38,380 --> 00:10:44,790
A times either the lambda one t
x one, and does that check out?
201
00:10:44,790 --> 00:10:47,550
Do we have equality there?
202
00:10:47,550 --> 00:10:52,190
Yes, because either the lambda
one t appears on both sides
203
00:10:52,190 --> 00:10:58,030
and the other one is Ax1
equal lambda one x1 -- check.
204
00:10:58,030 --> 00:11:02,590
Do you -- so, the -- we've come
to the first point to remember.
205
00:11:02,590 --> 00:11:05,030
These pure solutions.
206
00:11:05,030 --> 00:11:10,760
Those pure solutions are the
-- those pure exponentials are
207
00:11:10,760 --> 00:11:14,280
the differential
equations analogue of --
208
00:11:14,280 --> 00:11:17,040
last time we had pure powers.
209
00:11:17,040 --> 00:11:19,540
Last time -- so --
210
00:11:19,540 --> 00:11:24,200
so last time, the
analog was lambda --
211
00:11:24,200 --> 00:11:29,020
lambda one to the K-th power
x1, some amount of that,
212
00:11:29,020 --> 00:11:35,130
plus some amount of lambda
two to the K-th power x2.
213
00:11:35,130 --> 00:11:37,590
That was our formula
from last time.
214
00:11:37,590 --> 00:11:41,690
I put it up just to -- so
your eye compares those two
215
00:11:41,690 --> 00:11:42,880
formulas.
216
00:11:42,880 --> 00:11:45,550
Powers of lambda in the --
217
00:11:45,550 --> 00:11:48,610
in the difference equation
-- that -- this was in the --
218
00:11:48,610 --> 00:11:55,480
this was for the equation
uk plus one equals A uk.
219
00:11:55,480 --> 00:11:59,890
That was for the finite
step -- stepping by one.
220
00:11:59,890 --> 00:12:02,330
And we got powers,
now this is the one
221
00:12:02,330 --> 00:12:04,920
we're interested in,
the exponentials.
222
00:12:04,920 --> 00:12:08,750
So -- so that's --
that's the solution --
223
00:12:08,750 --> 00:12:10,660
what are c1 and c2?
224
00:12:10,660 --> 00:12:12,300
Then we're through.
225
00:12:12,300 --> 00:12:14,060
What are c1 and c2?
226
00:12:14,060 --> 00:12:17,340
Well, of course we know
these actual things.
227
00:12:17,340 --> 00:12:22,260
Let me just -- let
me come back to this.
228
00:12:22,260 --> 00:12:26,790
c1 is -- we haven't figured out
yet, but e to the lambda one t,
229
00:12:26,790 --> 00:12:32,890
the lambda one is zero so that's
just a one times x1 which is
230
00:12:32,890 --> 00:12:34,260
two one.
231
00:12:34,260 --> 00:12:39,590
So it's c1 times this one that's
not moving times the vector,
232
00:12:39,590 --> 00:12:44,660
the eigenvector two
one and c2 times --
233
00:12:44,660 --> 00:12:47,730
what's e to the lambda two t?
234
00:12:47,730 --> 00:12:51,830
Lambda two is minus three.
235
00:12:51,830 --> 00:12:54,010
So this is the term
that has the minus
236
00:12:54,010 --> 00:12:58,360
three t and its eigenvector
is this one minus one.
237
00:13:01,520 --> 00:13:06,640
So this vector
solves the equation
238
00:13:06,640 --> 00:13:08,580
and any multiple of it.
239
00:13:08,580 --> 00:13:12,190
This vector solves the equation
if it's got that factor
240
00:13:12,190 --> 00:13:14,620
e to the minus three t.
241
00:13:14,620 --> 00:13:17,980
We've got the answer
except for c1 and c2.
242
00:13:17,980 --> 00:13:22,970
So -- so everything I've done
is immediate as soon as you know
243
00:13:22,970 --> 00:13:25,570
the eigenvalues
and eigenvectors.
244
00:13:25,570 --> 00:13:27,640
So how do we get c1 and c2?
245
00:13:27,640 --> 00:13:30,930
That has to come from
the initial condition.
246
00:13:30,930 --> 00:13:38,740
So now I -- now I use -- u
of zero is given as one zero.
247
00:13:41,780 --> 00:13:46,346
So this is the initial condition
that will find c1 and c2.
248
00:13:46,346 --> 00:13:48,095
So let me do that on
the board underneath.
249
00:13:51,080 --> 00:13:52,935
At t equals zero, then --
250
00:13:56,500 --> 00:14:05,280
I get c1 times this guy plus
c2 and now I'm at times zero.
251
00:14:05,280 --> 00:14:08,320
So that's a one and
this is a one minus one
252
00:14:08,320 --> 00:14:12,920
and that's supposed to agree
with u of zero one zero.
253
00:14:19,550 --> 00:14:20,850
Okay.
254
00:14:20,850 --> 00:14:23,090
That should be two equations.
255
00:14:23,090 --> 00:14:26,500
That should give me c1 and
c2 and then I'm through.
256
00:14:26,500 --> 00:14:28,540
So what are c1 and c2?
257
00:14:28,540 --> 00:14:30,430
Let's see.
258
00:14:30,430 --> 00:14:33,000
I guess we could
actually spot them by eye
259
00:14:33,000 --> 00:14:36,800
or we could solve two
equations in two unknowns.
260
00:14:36,800 --> 00:14:38,090
Let's see.
261
00:14:38,090 --> 00:14:40,940
If these were both ones
-- so I'm just adding --
262
00:14:40,940 --> 00:14:43,630
then I would get three zero.
263
00:14:43,630 --> 00:14:46,740
So what's the -- what's
the solution, then?
264
00:14:49,970 --> 00:14:53,910
If -- if c1 and c2 are both
ones, I get three zero,
265
00:14:53,910 --> 00:14:55,990
so I want, like,
one third of that,
266
00:14:55,990 --> 00:14:57,750
because I want to get one zero.
267
00:14:57,750 --> 00:15:02,220
So I think it's c1 equals
a third, c2 equals a third.
268
00:15:05,460 --> 00:15:08,030
So finally I have the answer.
269
00:15:08,030 --> 00:15:11,000
Let me keep it in the
-- in this board here.
270
00:15:11,000 --> 00:15:20,530
Finally the answer is one third
of this plus one third of this.
271
00:15:24,450 --> 00:15:27,990
Do you see what -- what's
actually happening with this
272
00:15:27,990 --> 00:15:28,880
flow?
273
00:15:28,880 --> 00:15:32,630
This flow started out at --
the solution started out at one
274
00:15:32,630 --> 00:15:34,140
zero.
275
00:15:34,140 --> 00:15:36,790
Started at one zero.
276
00:15:36,790 --> 00:15:41,290
Then as time went on,
people moved, essentially.
277
00:15:41,290 --> 00:15:46,190
Some fraction of
this one moved here.
278
00:15:46,190 --> 00:15:52,030
And -- and in the limit, there's
-- there's the limit, as --
279
00:15:52,030 --> 00:15:52,530
right?
280
00:15:52,530 --> 00:15:55,540
As t goes to infinity,
as t gets very large,
281
00:15:55,540 --> 00:15:59,110
this disappears and this
is the steady state.
282
00:15:59,110 --> 00:16:02,550
So the steady state is --
283
00:16:02,550 --> 00:16:04,272
so the steady state --
284
00:16:08,700 --> 00:16:14,190
u -- we could call it u at
infinity is one third of two
285
00:16:14,190 --> 00:16:15,040
and one.
286
00:16:15,040 --> 00:16:17,110
It's -- it's two
thirds of one third.
287
00:16:19,970 --> 00:16:22,790
So that's the -- we really --
288
00:16:22,790 --> 00:16:25,280
I mean, you're
getting, like, total,
289
00:16:25,280 --> 00:16:29,790
insight into the
behavior of the solution,
290
00:16:29,790 --> 00:16:32,050
what the differential
equation does.
291
00:16:32,050 --> 00:16:37,480
Of course, we don't -- wouldn't
always have a steady state.
292
00:16:37,480 --> 00:16:40,580
Sometimes we would
approach zero.
293
00:16:40,580 --> 00:16:42,320
Sometimes we would blow up.
294
00:16:42,320 --> 00:16:45,250
Can we straighten
out those cases?
295
00:16:45,250 --> 00:16:47,510
The eigenvalue should tell us.
296
00:16:47,510 --> 00:16:50,170
So when do we get --
297
00:16:50,170 --> 00:16:54,440
so -- so let me ask first,
when do we get stability?
298
00:16:57,220 --> 00:17:00,250
That's u of t going to zero.
299
00:17:03,070 --> 00:17:05,660
When would the solution
go to zero no matter
300
00:17:05,660 --> 00:17:09,230
what the initial condition is?
301
00:17:09,230 --> 00:17:11,140
Negative eigenvalues, right.
302
00:17:11,140 --> 00:17:12,609
Negative eigenvalues.
303
00:17:12,609 --> 00:17:13,630
But now I have to --
304
00:17:13,630 --> 00:17:16,950
I have to ask you
for one more step.
305
00:17:16,950 --> 00:17:20,420
Suppose the eigenvalues
are complex numbers?
306
00:17:20,420 --> 00:17:22,680
Because we know they could be.
307
00:17:22,680 --> 00:17:27,760
Then we want stability --
this -- this -- we want --
308
00:17:27,760 --> 00:17:35,260
we need all these e to the
lambda t-s all going to zero
309
00:17:35,260 --> 00:17:40,920
and somehow that asks us
to have lambda negative.
310
00:17:40,920 --> 00:17:43,470
But suppose lambda
is a complex number?
311
00:17:43,470 --> 00:17:45,690
Then what's the test?
312
00:17:45,690 --> 00:17:50,340
What -- if lambda's a
complex number like, oh,
313
00:17:50,340 --> 00:17:54,730
suppose lambda is negative
plus an imaginary part?
314
00:17:54,730 --> 00:17:59,810
Say lambda is minus
three plus six i?
315
00:17:59,810 --> 00:18:01,120
What -- what happens then?
316
00:18:01,120 --> 00:18:03,530
Can we just, like,
do a -- a case here?
317
00:18:03,530 --> 00:18:11,550
If -- if this lambda is
minus three plus six it,
318
00:18:11,550 --> 00:18:14,170
how big is that number?
319
00:18:14,170 --> 00:18:18,450
Does this -- does this imaginary
part play a -- play a --
320
00:18:18,450 --> 00:18:20,840
play a role here or not?
321
00:18:20,840 --> 00:18:22,850
Or how big is --
322
00:18:22,850 --> 00:18:25,700
what's the absolute value
of that -- of that quantity?
323
00:18:28,530 --> 00:18:32,670
It's just e to the
minus three t, right?
324
00:18:32,670 --> 00:18:36,880
Because this other part, this --
the -- the magnitude -- the --
325
00:18:36,880 --> 00:18:41,795
this -- e to the six it -- what
-- that has absolute value one.
326
00:18:44,680 --> 00:18:45,180
Right?
327
00:18:45,180 --> 00:18:50,540
That's just this cosine of
six t plus i, sine of six t.
328
00:18:50,540 --> 00:18:53,060
And the absolute
value squared will
329
00:18:53,060 --> 00:18:56,230
be cos squared plus sine
squared will be one.
330
00:18:56,230 --> 00:18:59,680
This is -- this complex number
is running around the unit
331
00:18:59,680 --> 00:19:00,660
circle.
332
00:19:00,660 --> 00:19:04,770
This com- this -- the -- it's
the real part that matters.
333
00:19:04,770 --> 00:19:07,020
This is what I'm trying to do.
334
00:19:07,020 --> 00:19:10,980
Real part of lambda
has to be negative.
335
00:19:10,980 --> 00:19:14,880
If lambda's a complex
number, it's the real part,
336
00:19:14,880 --> 00:19:19,200
the minus three, that
either makes us go to zero
337
00:19:19,200 --> 00:19:24,940
or doesn't, or let
-- or blows up.
338
00:19:24,940 --> 00:19:27,380
The imaginary part won't
-- will just, like,
339
00:19:27,380 --> 00:19:30,690
oscillate between
the two components.
340
00:19:30,690 --> 00:19:31,360
Okay.
341
00:19:31,360 --> 00:19:33,230
So that's stability.
342
00:19:33,230 --> 00:19:36,040
And what about --
343
00:19:36,040 --> 00:19:37,305
what about a steady state?
344
00:19:42,130 --> 00:19:45,490
When would we have,
a steady state,
345
00:19:45,490 --> 00:19:47,390
always in the same direction?
346
00:19:47,390 --> 00:19:48,160
So let me --
347
00:19:48,160 --> 00:19:51,280
I'll take this part away --
348
00:19:51,280 --> 00:19:54,280
when -- so that was, like,
checking that it's --
349
00:19:54,280 --> 00:19:57,830
that it's the real part
that we care about.
350
00:19:57,830 --> 00:20:01,530
Now, we have a
steady state when --
351
00:20:01,530 --> 00:20:12,500
when lambda one is zero and the
other eigenvalues have what?
352
00:20:12,500 --> 00:20:14,990
So I'm looking -- like,
that example was, like,
353
00:20:14,990 --> 00:20:18,910
perfect for a steady state.
354
00:20:18,910 --> 00:20:22,760
We have a zero eigenvalue
and the other eigenvalues,
355
00:20:22,760 --> 00:20:25,070
we want those to disappear.
356
00:20:25,070 --> 00:20:28,975
So the other eigenvalues
have real part negative.
357
00:20:31,880 --> 00:20:35,700
And we blow up, for sure --
358
00:20:35,700 --> 00:20:45,380
we blow up if any real
part of lambda is positive.
359
00:20:49,090 --> 00:20:54,240
So if I -- if I reverse the
sign of A -- of the matrix A --
360
00:20:54,240 --> 00:20:57,420
suppose instead of the matrix
I had, the A that I had,
361
00:20:57,420 --> 00:20:58,390
I changed it --
362
00:20:58,390 --> 00:21:00,770
I changed all its sines.
363
00:21:00,770 --> 00:21:04,950
What would that do to the
eigenvalues and eigenvectors?
364
00:21:04,950 --> 00:21:08,090
If I -- if -- if I know the
eigenvalues and eigenvectors
365
00:21:08,090 --> 00:21:11,520
of A, tell me about minus A.
366
00:21:11,520 --> 00:21:14,780
What happens to the eigenvalues?
367
00:21:14,780 --> 00:21:18,410
For minus A, they'll
all change sine.
368
00:21:18,410 --> 00:21:20,660
So I'll have blow up.
369
00:21:20,660 --> 00:21:23,020
This -- instead of the
e to the minus three t,
370
00:21:23,020 --> 00:21:26,460
if I change that to minus --
if I change the sines in that
371
00:21:26,460 --> 00:21:30,810
matrix, I would change
the trace to plus three,
372
00:21:30,810 --> 00:21:34,020
I would have an e to the plus
three t and I would have blow
373
00:21:34,020 --> 00:21:36,150
up.
374
00:21:36,150 --> 00:21:39,430
Of course the zero eigenvalue
would stay at zero,
375
00:21:39,430 --> 00:21:42,490
but the other guy
is taking off in --
376
00:21:42,490 --> 00:21:45,091
if I reversed all the sines.
377
00:21:45,091 --> 00:21:45,590
Okay.
378
00:21:45,590 --> 00:21:51,090
So this is -- this is
the stability picture.
379
00:21:51,090 --> 00:21:56,680
And for a two by two
matrix, we can actually
380
00:21:56,680 --> 00:22:01,220
pin down even more
closely what that means.
381
00:22:01,220 --> 00:22:02,710
Can I -- let -- can I do that?
382
00:22:02,710 --> 00:22:04,410
Let me do that --
383
00:22:04,410 --> 00:22:05,810
I want to --
384
00:22:05,810 --> 00:22:11,230
for a two by two matrix, I
can tell whether the real part
385
00:22:11,230 --> 00:22:14,740
of the eigenvalues is
negative, I -- well, let me --
386
00:22:14,740 --> 00:22:18,480
let me tell you what I
have in mind for that.
387
00:22:18,480 --> 00:22:21,040
So two by two stability --
388
00:22:21,040 --> 00:22:25,750
let me -- this is
just a little comment.
389
00:22:25,750 --> 00:22:27,506
Two by two stability.
390
00:22:31,240 --> 00:22:35,930
So my matrix, now,
is just a b c d
391
00:22:35,930 --> 00:22:41,770
and I'm looking for the real
parts of both eigenvalues
392
00:22:41,770 --> 00:22:42,910
to be negative.
393
00:22:47,480 --> 00:22:47,980
Okay.
394
00:22:52,330 --> 00:22:55,300
What -- how can I tell
from looking at the matrix,
395
00:22:55,300 --> 00:22:58,230
without computing
its eigenvalues,
396
00:22:58,230 --> 00:23:02,150
whether the two
eigenvalues are negative,
397
00:23:02,150 --> 00:23:04,930
or at least their real
parts are negative?
398
00:23:04,930 --> 00:23:07,260
What would that tell
me about the trace?
399
00:23:07,260 --> 00:23:10,830
So -- so the trace --
400
00:23:10,830 --> 00:23:14,930
that's this a plus d --
401
00:23:14,930 --> 00:23:19,470
what can you tell me about
the trace in the case of a two
402
00:23:19,470 --> 00:23:21,760
by two stable matrix?
403
00:23:21,760 --> 00:23:25,320
That means the eigenvalues
have -- are negative,
404
00:23:25,320 --> 00:23:28,660
or at least the real parts of
those eigenvalues are negative
405
00:23:28,660 --> 00:23:33,140
-- then, when I take the -- when
I look at the matrix and find
406
00:23:33,140 --> 00:23:36,930
its trace, what -- what
do I know about that?
407
00:23:36,930 --> 00:23:38,360
It's negative, right.
408
00:23:38,360 --> 00:23:40,940
This is the sum of
the -- this equals --
409
00:23:40,940 --> 00:23:47,010
this equals lambda one plus
lambda two, so it's negative.
410
00:23:47,010 --> 00:23:49,590
The two eigenvalues, by
the way, will have --
411
00:23:49,590 --> 00:23:54,990
if they're complex -- will have
plus six i and minus six i.
412
00:23:54,990 --> 00:23:59,860
The complex parts will -- will
be conjugates of each other
413
00:23:59,860 --> 00:24:04,720
and disappear when we add and
the trace will be negative.
414
00:24:04,720 --> 00:24:06,710
Okay, the trace
has to be negative.
415
00:24:06,710 --> 00:24:09,030
Is that enough --
416
00:24:09,030 --> 00:24:14,670
is a negative trace enough
to make the matrix stable?
417
00:24:14,670 --> 00:24:16,180
Shouldn't be enough, right?
418
00:24:16,180 --> 00:24:19,270
Can I -- can you make -- what's
a matrix that has a negative
419
00:24:19,270 --> 00:24:24,040
trace but still it's not stable?
420
00:24:24,040 --> 00:24:27,500
So it -- it has a blow -- it
still has a blow-up factor
421
00:24:27,500 --> 00:24:30,900
and a -- and a --
and a decaying one.
422
00:24:30,900 --> 00:24:33,820
So what would be a -- so
just -- just to see --
423
00:24:33,820 --> 00:24:35,920
maybe I just put that here.
424
00:24:35,920 --> 00:24:40,790
This -- now I'm looking for an
example where the trace could
425
00:24:40,790 --> 00:24:48,080
be negative but still blow up.
426
00:24:48,080 --> 00:24:52,830
Of course -- yeah,
let's just take one.
427
00:24:52,830 --> 00:24:57,990
Oh, look, let me -- let me make
it minus two zero zero one.
428
00:24:57,990 --> 00:25:00,140
Okay.
429
00:25:00,140 --> 00:25:04,810
There's a case where that
matrix has negative trace --
430
00:25:04,810 --> 00:25:06,390
I know its
eigenvalues of course.
431
00:25:06,390 --> 00:25:09,750
They're minus two and
one and it blows up.
432
00:25:09,750 --> 00:25:12,780
It's got -- it's got a
plus one eigenvalue here,
433
00:25:12,780 --> 00:25:17,280
so there would be an e to
the plus t in the solution
434
00:25:17,280 --> 00:25:21,170
and it'll blow up if it has
any second component at all.
435
00:25:21,170 --> 00:25:23,900
I need another condition.
436
00:25:23,900 --> 00:25:25,615
And it's a condition
on the determinant.
437
00:25:28,240 --> 00:25:29,560
And what's that condition?
438
00:25:29,560 --> 00:25:32,730
If I know that the
two eigenvalues --
439
00:25:32,730 --> 00:25:36,090
suppose I know they're
negative, both negative.
440
00:25:36,090 --> 00:25:39,790
What does that tell me
about the determinant?
441
00:25:39,790 --> 00:25:41,260
Let me ask again.
442
00:25:41,260 --> 00:25:44,990
If I know both the
eigenvalues are negative,
443
00:25:44,990 --> 00:25:47,760
then I know the
trace is negative
444
00:25:47,760 --> 00:25:53,170
but the determinant is
positive, because it's
445
00:25:53,170 --> 00:25:56,450
the product of the
two eigenvalues.
446
00:25:56,450 --> 00:26:00,580
So this determinant is
lambda one times lambda two.
447
00:26:00,580 --> 00:26:04,540
This is -- this is lambda
one times lambda two
448
00:26:04,540 --> 00:26:08,060
and if they're both negative,
the product is positive.
449
00:26:08,060 --> 00:26:11,860
So positive determinant,
negative trace.
450
00:26:11,860 --> 00:26:17,200
I can easily track down the --
this condition also for the --
451
00:26:17,200 --> 00:26:20,380
if -- if there's an imaginary
part hanging around.
452
00:26:20,380 --> 00:26:20,880
Okay.
453
00:26:20,880 --> 00:26:25,550
So that's a -- like a
small but quite useful,
454
00:26:25,550 --> 00:26:29,820
because two by two is
always important --
455
00:26:29,820 --> 00:26:33,100
comment on stability.
456
00:26:33,100 --> 00:26:33,750
Okay.
457
00:26:33,750 --> 00:26:40,170
So let's just look
at the picture again.
458
00:26:40,170 --> 00:26:41,290
Okay.
459
00:26:41,290 --> 00:26:43,980
The main part of my
lecture, the one thing
460
00:26:43,980 --> 00:26:46,700
you want to be able to,
like, just do automatically
461
00:26:46,700 --> 00:26:51,750
is this step of
solving the system.
462
00:26:51,750 --> 00:26:54,190
Find the eigenvalues,
find the eigenvectors,
463
00:26:54,190 --> 00:26:56,000
find the coefficients.
464
00:26:56,000 --> 00:27:01,090
And notice -- what's the matrix
-- in this linear system here,
465
00:27:01,090 --> 00:27:05,590
I can't help -- if I -- if I
have equations like that --
466
00:27:05,590 --> 00:27:08,900
that's my equations
column at a time --
467
00:27:08,900 --> 00:27:11,710
what's the matrix
form of that equation?
468
00:27:11,710 --> 00:27:18,580
So -- so this -- this
system of equations is --
469
00:27:18,580 --> 00:27:26,710
is some matrix multiplying
c1, c2 to give u of zero.
470
00:27:26,710 --> 00:27:29,370
One zero.
471
00:27:29,370 --> 00:27:30,510
What's the matrix?
472
00:27:30,510 --> 00:27:35,400
Well, it's obviously
two one, one minus one,
473
00:27:35,400 --> 00:27:37,760
but we have a name, or
at least a letter --
474
00:27:37,760 --> 00:27:40,090
actually a name for that matrix.
475
00:27:40,090 --> 00:27:42,670
Wh- what matrix
are we s- are we --
476
00:27:42,670 --> 00:27:47,390
are we dealing with here in
this step of finding the c-s?
477
00:27:47,390 --> 00:27:50,690
Its letter is S --
478
00:27:50,690 --> 00:27:52,340
it's the eigenvector matrix.
479
00:27:52,340 --> 00:27:52,970
Of course.
480
00:27:52,970 --> 00:27:55,230
These are the
eigenvectors, there
481
00:27:55,230 --> 00:27:57,150
in the columns of our matrix.
482
00:27:57,150 --> 00:28:02,520
So this is S c
equals u of zero --
483
00:28:02,520 --> 00:28:09,640
is the -- that step where you
find the actual coefficients,
484
00:28:09,640 --> 00:28:14,690
you find out how much of
each pure exponential is
485
00:28:14,690 --> 00:28:16,910
in the solution.
486
00:28:16,910 --> 00:28:20,660
By getting it right at the
start, then it stays right
487
00:28:20,660 --> 00:28:21,320
forever.
488
00:28:21,320 --> 00:28:24,820
I -- you're seeing this
picture that each --
489
00:28:24,820 --> 00:28:29,090
each pure exponential goes on
its own way once you start it
490
00:28:29,090 --> 00:28:29,620
from u of
491
00:28:29,620 --> 00:28:30,490
zero.
492
00:28:30,490 --> 00:28:33,810
So you start it by
figuring out how much
493
00:28:33,810 --> 00:28:37,880
each one is present in u of
zero and then off they go.
494
00:28:37,880 --> 00:28:38,740
Okay.
495
00:28:38,740 --> 00:28:45,110
So -- so that's a system of
two equations in two unknowns
496
00:28:45,110 --> 00:28:48,070
coupled --
497
00:28:48,070 --> 00:28:53,630
the matrix sort of couples
u1 and u2 and the eigenvalues
498
00:28:53,630 --> 00:28:57,770
and eigenvectors uncouple
it, diagonalize it.
499
00:28:57,770 --> 00:29:00,830
Actually -- okay, now can I --
500
00:29:00,830 --> 00:29:04,600
can I think in terms
of S and lambda?
501
00:29:04,600 --> 00:29:07,330
So I want to write
the solution down,
502
00:29:07,330 --> 00:29:10,840
again in terms of S and lambda.
503
00:29:10,840 --> 00:29:11,340
Okay.
504
00:29:11,340 --> 00:29:14,890
I'll do that on this far board.
505
00:29:14,890 --> 00:29:15,890
Okay.
506
00:29:15,890 --> 00:29:20,540
So I'm coming back to --
507
00:29:20,540 --> 00:29:26,470
I'm coming back to our
equation du/dt equals Au.
508
00:29:26,470 --> 00:29:35,890
Now this matrix A couples them.
509
00:29:35,890 --> 00:29:39,010
The whole point of
eigenvectors is to uncouple.
510
00:29:39,010 --> 00:29:46,510
So one way to see that is
introduce set u equal A --
511
00:29:46,510 --> 00:29:47,010
not
512
00:29:47,010 --> 00:29:54,580
A. It's S, the eigenvector
matrix that uncouples it.
513
00:29:54,580 --> 00:29:58,890
So I'm -- I'm taking this
equation as I'm given,
514
00:29:58,890 --> 00:30:04,150
coupled with -- with A has --
is probably full of non-zeroes,
515
00:30:04,150 --> 00:30:08,130
but I'm -- by uncoupling it,
I mean I'm diagonalizing it.
516
00:30:08,130 --> 00:30:11,630
If I can get a
diagonal matrix, I'm --
517
00:30:11,630 --> 00:30:12,470
I'm in.
518
00:30:12,470 --> 00:30:13,120
Okay.
519
00:30:13,120 --> 00:30:14,950
So I plug that in.
520
00:30:14,950 --> 00:30:18,720
This is A S v.
521
00:30:18,720 --> 00:30:20,813
And this is S dv/dt.
522
00:30:25,810 --> 00:30:26,890
S is a constant.
523
00:30:26,890 --> 00:30:30,050
It's -- this it the
eigenvector matrix.
524
00:30:30,050 --> 00:30:31,635
This is the eigenvector matrix.
525
00:30:34,820 --> 00:30:35,330
Okay.
526
00:30:35,330 --> 00:30:37,550
Now I'm going to bring
S inverse over here.
527
00:30:40,390 --> 00:30:41,285
And what have I got?
528
00:30:45,160 --> 00:30:53,470
That combination S inverse A S
is lambda, the diagonal matrix.
529
00:30:53,470 --> 00:30:56,940
That's -- that's the
point, that in --
530
00:30:56,940 --> 00:31:01,400
if I'm using the
eigenvectors as my basis,
531
00:31:01,400 --> 00:31:06,700
then my system of
equations is just diagonal.
532
00:31:06,700 --> 00:31:10,420
I -- each -- there's
no coupling anymore --
533
00:31:10,420 --> 00:31:13,130
dv1/dt is lambda one v1.
534
00:31:13,130 --> 00:31:20,890
So let's just write that
down. dv1/ dt is lambda one v1
535
00:31:20,890 --> 00:31:26,470
and so on for all
n of the equations.
536
00:31:26,470 --> 00:31:29,610
It's a system of equations
but they're not connected,
537
00:31:29,610 --> 00:31:32,865
so they're easy to solve
and why don't I just
538
00:31:32,865 --> 00:31:33,865
write down the solution?
539
00:31:36,880 --> 00:31:47,160
v -- well, v is now some
e to the lambda one t --
540
00:31:47,160 --> 00:31:49,600
well, okay --
541
00:31:49,600 --> 00:31:56,330
I guess my idea here now is to
use, the natural notation --
542
00:31:56,330 --> 00:32:04,740
v of T is e to the
lambda tv of zero.
543
00:32:04,740 --> 00:32:16,150
And u of t will be Se to the
lambda t S inverse, u of zero.
544
00:32:16,150 --> 00:32:20,990
This is the -- this is the,
formula I'm headed for.
545
00:32:25,220 --> 00:32:29,530
This -- this matrix, S e
to the lambda t S inverse,
546
00:32:29,530 --> 00:32:32,040
that's my exponential.
547
00:32:32,040 --> 00:32:40,483
That's my e to the A t, is this
S e to the lambda t S inverse.
548
00:32:43,600 --> 00:32:47,310
So my -- my job really now is
to explain what's going on with
549
00:32:47,310 --> 00:32:49,620
this matrix up in
the exponential.
550
00:32:49,620 --> 00:32:51,610
What does that mean?
551
00:32:51,610 --> 00:32:54,110
What does it mean to
say e to a matrix?
552
00:32:58,100 --> 00:33:01,680
This ought to be easier because
this is e to a diagonal matrix,
553
00:33:01,680 --> 00:33:03,970
but still it's a matrix.
554
00:33:03,970 --> 00:33:07,350
What do we mean by e to the A t?
555
00:33:07,350 --> 00:33:13,120
Because really e to the
A t is my answer here.
556
00:33:13,120 --> 00:33:18,620
My -- my answer to
this equation is --
557
00:33:18,620 --> 00:33:26,170
this u of t, this is my -- this
is my e to the A t u of zero.
558
00:33:26,170 --> 00:33:30,907
So it's -- my job is
really now to say what's --
559
00:33:30,907 --> 00:33:31,740
what does that mean?
560
00:33:31,740 --> 00:33:33,780
What's the exponential
of a matrix
561
00:33:33,780 --> 00:33:38,390
and why is that formula correct?
562
00:33:38,390 --> 00:33:38,950
Okay.
563
00:33:38,950 --> 00:33:42,190
So I'll put that on
the board underneath.
564
00:33:42,190 --> 00:33:45,210
What's the exponential
of a matrix?
565
00:33:45,210 --> 00:33:47,430
Let me come back here.
566
00:33:47,430 --> 00:33:49,460
So I'm talking about
matrix exponentials.
567
00:33:55,040 --> 00:33:57,540
e to the At.
568
00:33:57,540 --> 00:33:58,350
Okay.
569
00:33:58,350 --> 00:34:01,090
How are we going to define
the exponential of a --
570
00:34:01,090 --> 00:34:01,735
of something?
571
00:34:04,490 --> 00:34:09,940
The trick -- the idea is --
the thing to go back to is
572
00:34:09,940 --> 00:34:15,860
exponential -- there's a
power series for exponentials.
573
00:34:15,860 --> 00:34:19,690
That's how you -- you -- the
good way to define e to the x
574
00:34:19,690 --> 00:34:25,469
is the power series one plus
x plus one half x squared,
575
00:34:25,469 --> 00:34:29,489
one six x cubed and we'll
do it now when the --
576
00:34:29,489 --> 00:34:30,770
when we have a matrix.
577
00:34:30,770 --> 00:34:34,739
So the one becomes the
identity, the x is At,
578
00:34:34,739 --> 00:34:42,620
the x squared is At squared
and I divide by two.
579
00:34:42,620 --> 00:34:47,600
The cube, the x cube
is At cubed over six,
580
00:34:47,600 --> 00:34:50,880
and what's the
general term in here?
581
00:34:50,880 --> 00:34:55,929
At to the n-th
power divided by --
582
00:34:55,929 --> 00:34:57,710
and it goes on.
583
00:34:57,710 --> 00:35:01,430
But what do I divide by?
584
00:35:01,430 --> 00:35:05,350
So, you see the pattern
-- here I divided by one,
585
00:35:05,350 --> 00:35:10,080
here I divided by one by two by
six, those are the factorials.
586
00:35:10,080 --> 00:35:10,965
It's n factorial.
587
00:35:14,110 --> 00:35:17,445
That was, like, the one
beautiful Taylor series.
588
00:35:20,300 --> 00:35:23,180
The one beautiful Taylor series
-- well, there are two --
589
00:35:23,180 --> 00:35:25,960
there are two beautiful
Taylor series in this world.
590
00:35:25,960 --> 00:35:29,550
The Taylor series
for e to the x is
591
00:35:29,550 --> 00:35:35,090
the s with x to the
n-th over n factorial.
592
00:35:35,090 --> 00:35:38,680
And all I'm doing is doing
the same thing for matrixes.
593
00:35:38,680 --> 00:35:40,850
The other beautiful
Taylor series
594
00:35:40,850 --> 00:35:47,920
is just the sum of x to the
n-th not divided by n factorial.
595
00:35:47,920 --> 00:35:51,300
Can you -- do you know
what function that one is?
596
00:35:51,300 --> 00:35:53,990
So if I take --
this is the series,
597
00:35:53,990 --> 00:35:58,070
all these sums are going
from zero to infinity.
598
00:35:58,070 --> 00:35:59,960
What's -- what
function have I got --
599
00:35:59,960 --> 00:36:02,770
this is like a side comment --
600
00:36:02,770 --> 00:36:06,750
this is one plus x plus x
squared plus x cubed plus x
601
00:36:06,750 --> 00:36:09,430
to the fourth not divided
by anything, what's --
602
00:36:09,430 --> 00:36:11,800
what's that function?
603
00:36:11,800 --> 00:36:15,380
One plus x plus x squared plus
x cubed plus x fourth forever
604
00:36:15,380 --> 00:36:18,700
is one over one minus x.
605
00:36:18,700 --> 00:36:24,120
It's the geometric series, the
nicest power series of all.
606
00:36:24,120 --> 00:36:28,850
So, actually, of course, there
would be a similar thing here.
607
00:36:28,850 --> 00:36:36,810
If -- if I wanted, I minus
A t inverse would be --
608
00:36:36,810 --> 00:36:39,060
now I've got matrixes.
609
00:36:39,060 --> 00:36:43,150
I've got matrixes everywhere,
but it's just like that series
610
00:36:43,150 --> 00:36:46,690
with -- and just like this
one without the divisions.
611
00:36:46,690 --> 00:36:56,310
It's I plus At plus At squared
plus At cubed and forever.
612
00:36:59,200 --> 00:37:02,460
So that's actually a --
a reasonable way to find
613
00:37:02,460 --> 00:37:04,660
the inverse of a matrix.
614
00:37:04,660 --> 00:37:07,500
If I chop it off --
615
00:37:07,500 --> 00:37:10,120
well, it's reasonable
if t is small.
616
00:37:10,120 --> 00:37:13,330
If t is a small number, then --
617
00:37:13,330 --> 00:37:15,940
then t squared is
extremely small,
618
00:37:15,940 --> 00:37:19,350
t cubed is even smaller,
so approximately
619
00:37:19,350 --> 00:37:22,480
that inverse is I plus At.
620
00:37:22,480 --> 00:37:24,660
I can keep more terms if I like.
621
00:37:24,660 --> 00:37:25,830
Do you see what I'm doing?
622
00:37:25,830 --> 00:37:31,440
I'm just saying we can do the
same thing for matrixes that we
623
00:37:31,440 --> 00:37:35,600
do for ordinary functions
and the good thing about
624
00:37:35,600 --> 00:37:38,470
the exponential
series -- so in a way,
625
00:37:38,470 --> 00:37:41,990
this series is
better than this one.
626
00:37:41,990 --> 00:37:43,290
Why?
627
00:37:43,290 --> 00:37:45,410
Because this one
always converges.
628
00:37:45,410 --> 00:37:48,440
I'm dividing by these
bigger and bigger numbers,
629
00:37:48,440 --> 00:37:54,770
so whatever matrix A and however
large t is, that series --
630
00:37:54,770 --> 00:37:57,430
these terms go to zero.
631
00:37:57,430 --> 00:38:01,960
The series adds up to a finite
sum, e to the At is a -- is --
632
00:38:01,960 --> 00:38:04,390
is completely defined.
633
00:38:04,390 --> 00:38:08,710
Whereas this second
guy could fail, right?
634
00:38:08,710 --> 00:38:11,730
If At is big --
635
00:38:11,730 --> 00:38:15,190
somehow if At has an
eigenvalue larger than one,
636
00:38:15,190 --> 00:38:18,690
then when I square it it'll
have that eigenvalue squared,
637
00:38:18,690 --> 00:38:21,820
when I cube it the
eigenvalue will be cubed --
638
00:38:21,820 --> 00:38:26,560
that series will blow up
unless the eigenvalues of At
639
00:38:26,560 --> 00:38:28,500
are smaller than one.
640
00:38:28,500 --> 00:38:32,150
So when the eigenvalues of
At are smaller than one --
641
00:38:32,150 --> 00:38:33,610
so I'd better put that in.
642
00:38:33,610 --> 00:38:38,260
The -- all eigenvalues
of At below one --
643
00:38:38,260 --> 00:38:42,230
then that series converges
and gives me the inverse.
644
00:38:42,230 --> 00:38:42,970
Okay.
645
00:38:42,970 --> 00:38:47,590
So this is the guy I'm chiefly
interested in, and I wanted
646
00:38:47,590 --> 00:38:51,820
to connect it to --
647
00:38:51,820 --> 00:38:52,400
oh, okay.
648
00:38:52,400 --> 00:38:55,810
What's -- how do I -- how do
I get -- this is my, like,
649
00:38:55,810 --> 00:38:58,160
main thing now to do --
650
00:38:58,160 --> 00:39:02,270
how do I get e to the At --
651
00:39:02,270 --> 00:39:05,760
how do I see that e to the
At is the same as this?
652
00:39:10,450 --> 00:39:16,410
In other words, I can find e to
the At by finding S and lambda,
653
00:39:16,410 --> 00:39:18,710
because now e to the lambda t
654
00:39:18,710 --> 00:39:21,350
is easy.
655
00:39:21,350 --> 00:39:24,820
Lambda's a diagonal matrix
and we can write down either
656
00:39:24,820 --> 00:39:27,250
the lambda t -- and will
right -- in a minute.
657
00:39:27,250 --> 00:39:29,810
But how -- do you see what --
658
00:39:29,810 --> 00:39:33,290
do you see that
we're hoping for a --
659
00:39:33,290 --> 00:39:38,540
we're hoping that we can
compute either the A T from S
660
00:39:38,540 --> 00:39:41,450
and lambda --
661
00:39:41,450 --> 00:39:44,980
and I have to look at that
definition and say, okay,
662
00:39:44,980 --> 00:39:48,660
how do -- how do I get an S and
the lambda to come out of that?
663
00:39:48,660 --> 00:39:50,670
Okay, can -- do you see how I --
664
00:39:50,670 --> 00:39:54,830
I want to connect that to
that, from that definition.
665
00:39:54,830 --> 00:39:59,090
So let me erase this --
the geometric series,
666
00:39:59,090 --> 00:40:08,250
which isn't part of the
differential equations case
667
00:40:08,250 --> 00:40:14,200
and get the S and the
lambda into this picture.
668
00:40:14,200 --> 00:40:15,900
Oh, okay.
669
00:40:15,900 --> 00:40:16,420
Here we go.
670
00:40:19,210 --> 00:40:22,930
So identity is fine.
671
00:40:22,930 --> 00:40:26,880
Now -- all right, you --
you -- you'll see how I'm --
672
00:40:26,880 --> 00:40:31,900
how I'm -- how I going to
get A replaced by S and S is
673
00:40:31,900 --> 00:40:32,550
in lambda's?
674
00:40:32,550 --> 00:40:36,520
Well I use the fundamental
formula of this whole chapter.
675
00:40:36,520 --> 00:40:43,540
A is S lambda S inverse
and then times t.
676
00:40:43,540 --> 00:40:45,451
That's At.
677
00:40:45,451 --> 00:40:45,950
Okay.
678
00:40:45,950 --> 00:40:48,910
What's A squared t?
679
00:40:48,910 --> 00:40:51,450
I can -- I've got
the divide by two,
680
00:40:51,450 --> 00:40:56,580
I've got the t squared
and I've got an A squared.
681
00:40:56,580 --> 00:41:02,580
All right, I -- so I've got
a -- there's A -- there's A.
682
00:41:02,580 --> 00:41:04,390
Now square it.
683
00:41:04,390 --> 00:41:05,880
So what happens
when I square it?
684
00:41:05,880 --> 00:41:08,050
We've seen that before.
685
00:41:08,050 --> 00:41:16,120
When I square it, I get S
lambda squared S inverse, right?
686
00:41:16,120 --> 00:41:20,070
When I square that thing,
the -- there's an S and a --
687
00:41:20,070 --> 00:41:23,840
an S cancels out an S inverse.
688
00:41:23,840 --> 00:41:25,900
I'm left with the S
on the left, the S
689
00:41:25,900 --> 00:41:29,140
inverse on the right and
lambda squared in the middle.
690
00:41:29,140 --> 00:41:31,660
And so on.
691
00:41:31,660 --> 00:41:36,680
The next one'll be S
lambda cubed, S inverse --
692
00:41:36,680 --> 00:41:39,590
times t cubed over
three factorial.
693
00:41:39,590 --> 00:41:45,190
And now -- what do I do now?
694
00:41:45,190 --> 00:41:48,440
I want to pull an S
out from everything.
695
00:41:48,440 --> 00:41:53,010
I want an S out of
the whole thing.
696
00:41:53,010 --> 00:41:57,020
Well, look, I'd better
write the identity how?
697
00:41:57,020 --> 00:42:01,250
I -- I want to be able to pull
an S out and an S inverse out
698
00:42:01,250 --> 00:42:04,840
from the other side, so I just
write the identity as S times S
699
00:42:04,840 --> 00:42:05,820
inverse.
700
00:42:05,820 --> 00:42:11,120
So I have an S there and
an S inverse from this side
701
00:42:11,120 --> 00:42:13,240
and what have I
got in the middle?
702
00:42:16,170 --> 00:42:18,241
If I pull out an S
and an S inverse,
703
00:42:18,241 --> 00:42:19,490
what have I got in the middle?
704
00:42:19,490 --> 00:42:23,260
I've got the
identity, a lambda t,
705
00:42:23,260 --> 00:42:26,650
a lambda squared t
squared over two --
706
00:42:26,650 --> 00:42:30,780
I've got e to the lambda t.
707
00:42:30,780 --> 00:42:32,600
That's what's in the middle.
708
00:42:32,600 --> 00:42:36,630
That's my formula
for e to the At.
709
00:42:36,630 --> 00:42:39,120
Oh, now I have to ask you.
710
00:42:39,120 --> 00:42:42,290
Does this formula always work?
711
00:42:42,290 --> 00:42:45,420
This formula always works --
712
00:42:45,420 --> 00:42:48,540
well, except it's
an infinite series.
713
00:42:48,540 --> 00:42:51,800
But what do I mean
by always work?
714
00:42:51,800 --> 00:42:55,300
And this one doesn't
always work and I just
715
00:42:55,300 --> 00:42:58,460
have to remind you
of what assumption
716
00:42:58,460 --> 00:43:00,660
is built into this
formula that's
717
00:43:00,660 --> 00:43:03,580
not built into the original.
718
00:43:03,580 --> 00:43:07,900
The assumption that A
can be diagonalized.
719
00:43:07,900 --> 00:43:11,660
You'll remember that
there are some small --
720
00:43:11,660 --> 00:43:14,770
sm- some subset of
matrixes that don't
721
00:43:14,770 --> 00:43:18,000
have n independent
eigenvectors, so we
722
00:43:18,000 --> 00:43:20,590
don't have an S inverse
for those matrixes
723
00:43:20,590 --> 00:43:24,780
and the whole
diagonalization breaks down.
724
00:43:24,780 --> 00:43:26,860
We could still
make it triangular.
725
00:43:26,860 --> 00:43:28,010
I'll tell you about that.
726
00:43:28,010 --> 00:43:32,930
But diagonal we can't do for
those particular degenerate
727
00:43:32,930 --> 00:43:37,310
matrixes that don't have enough
independent eigenvectors.
728
00:43:37,310 --> 00:43:40,240
But otherwise, this is golden.
729
00:43:40,240 --> 00:43:40,970
Okay.
730
00:43:40,970 --> 00:43:44,780
So that's the formula --
that's the matrix exponential.
731
00:43:44,780 --> 00:43:48,680
Now it just remains for me to
say what is e to the lambda t?
732
00:43:48,680 --> 00:43:50,460
Can I just do that?
733
00:43:50,460 --> 00:43:55,280
Let me just put that
in the corner here.
734
00:43:55,280 --> 00:44:02,180
What is the exponential
of a diagonal matrix?
735
00:44:02,180 --> 00:44:10,140
Remember lambda is diagonal,
lambda one down to lambda n.
736
00:44:10,140 --> 00:44:14,520
What's the exponential
of that diagonal matrix?
737
00:44:14,520 --> 00:44:19,550
Because our whole point is
that this ought to be simple.
738
00:44:19,550 --> 00:44:23,240
Our whole point is that to take
the exponential of a diagonal
739
00:44:23,240 --> 00:44:27,560
matrix ought to be
completely decoupled --
740
00:44:27,560 --> 00:44:30,300
it ought to be diagonal,
in other words, and it is.
741
00:44:30,300 --> 00:44:37,010
It's just e to the lambda
one t, e to the lambda two t,
742
00:44:37,010 --> 00:44:41,070
e to the lambda n t, all zeroes.
743
00:44:41,070 --> 00:44:47,620
So -- so if we have a diagonal
matrix and I plug it into this
744
00:44:47,620 --> 00:44:52,140
exponential formula,
everything's diagonal
745
00:44:52,140 --> 00:44:55,710
and the diagonal terms are
just the ordinary scaler
746
00:44:55,710 --> 00:44:58,930
exponentials e to
the lambda one t.
747
00:44:58,930 --> 00:45:01,840
Okay, so that's -- that's --
748
00:45:01,840 --> 00:45:06,050
in a sense, I'm doing here, on
this board, with -- with, like,
749
00:45:06,050 --> 00:45:11,100
formulas what I did on the --
750
00:45:11,100 --> 00:45:15,940
in the first half of the
lecture with specific matrix A
751
00:45:15,940 --> 00:45:19,270
and specific eigenvalues
and eigenvectors.
752
00:45:19,270 --> 00:45:21,990
The -- so let me show
you the formulas again.
753
00:45:21,990 --> 00:45:24,770
But the -- so you --
754
00:45:24,770 --> 00:45:27,300
I guess -- what should
you know from this
755
00:45:27,300 --> 00:45:28,300
lecture?
756
00:45:28,300 --> 00:45:34,180
You should know what this
matrix exponential is and, like,
757
00:45:34,180 --> 00:45:36,670
when does it go to zero?
758
00:45:36,670 --> 00:45:38,390
Tell me again, now,
the answer to that.
759
00:45:38,390 --> 00:45:41,350
When does e to
the At approach --
760
00:45:41,350 --> 00:45:45,510
get smaller and
smaller as t increases?
761
00:45:45,510 --> 00:45:49,070
Well, the S and the S
inverse aren't moving.
762
00:45:49,070 --> 00:45:51,780
It's this one that has to
get smaller and smaller
763
00:45:51,780 --> 00:45:57,160
and that one has this
simple diagonal form.
764
00:45:57,160 --> 00:46:01,740
And it goes to zero provided
every one of these lambdas --
765
00:46:01,740 --> 00:46:04,840
I -- I need to have each one
of these guys go to zero,
766
00:46:04,840 --> 00:46:09,930
so I need every real part of
every eigenvalue negative.
767
00:46:12,650 --> 00:46:13,230
Right?
768
00:46:13,230 --> 00:46:15,690
If the real part is
negative, that's --
769
00:46:15,690 --> 00:46:19,160
that takes the exponential
-- that forces --
770
00:46:19,160 --> 00:46:21,420
the exponential goes to zero.
771
00:46:21,420 --> 00:46:24,480
Okay, so that -- that's
really the difference.
772
00:46:24,480 --> 00:46:33,250
If I can just draw the -- here's
a picture of the -- of the --
773
00:46:33,250 --> 00:46:36,780
this is the complex plain.
774
00:46:36,780 --> 00:46:42,080
Here's the real axis and
here's the imaginary axis.
775
00:46:42,080 --> 00:46:43,960
And where do the
eigenvalues have
776
00:46:43,960 --> 00:46:47,570
to be for stability in
differential equations?
777
00:46:47,570 --> 00:46:52,470
They have to be over here,
in the left half plain.
778
00:46:52,470 --> 00:46:55,910
So the left half plain is this
plain, real part of lambda,
779
00:46:55,910 --> 00:46:58,820
less than zero.
780
00:46:58,820 --> 00:47:01,190
Where do the ma- where
do the eigenvalues have
781
00:47:01,190 --> 00:47:06,500
to be for powers of the
matrix to go to zero?
782
00:47:06,500 --> 00:47:11,960
Powers of the matrix go to zero
if the eigenvalues are in here.
783
00:47:11,960 --> 00:47:17,800
So this is the stability
region for powers --
784
00:47:17,800 --> 00:47:22,490
this is the region -- absolute
value of lambda, less than one.
785
00:47:22,490 --> 00:47:26,740
That's the stability for -- that
tells us that the powers of A
786
00:47:26,740 --> 00:47:30,260
go to zero, this tells us
that the exponential of A goes
787
00:47:30,260 --> 00:47:31,220
to zero.
788
00:47:31,220 --> 00:47:31,790
Okay.
789
00:47:31,790 --> 00:47:33,700
One final example.
790
00:47:33,700 --> 00:47:38,580
Let me just write down how
to deal with a final example.
791
00:47:38,580 --> 00:47:39,980
Let's see.
792
00:47:44,480 --> 00:47:48,930
So my final example will be a
single equation, u''+bu'+Ku=0.
793
00:47:57,040 --> 00:48:01,000
One equation, second order.
794
00:48:01,000 --> 00:48:03,490
How do I --
795
00:48:03,490 --> 00:48:05,290
and maybe I should have used --
796
00:48:05,290 --> 00:48:08,170
I'll use -- I prefer
to use y here,
797
00:48:08,170 --> 00:48:12,170
because that's what you see
in differential equations.
798
00:48:12,170 --> 00:48:14,790
And I want u to be a vector.
799
00:48:14,790 --> 00:48:23,730
So how do I change one second
order equation into a two
800
00:48:23,730 --> 00:48:28,250
by two first order system?
801
00:48:28,250 --> 00:48:30,540
Just the way I
did for Fibonacci.
802
00:48:30,540 --> 00:48:38,180
I'll let u be y prime and y.
803
00:48:38,180 --> 00:48:43,210
What I'm going to do is I'm
going to add an extra equation,
804
00:48:43,210 --> 00:48:46,620
y prime equals y prime.
805
00:48:46,620 --> 00:48:50,800
So I take this -- so by --
806
00:48:50,800 --> 00:48:55,110
so using this as
the vector unknown,
807
00:48:55,110 --> 00:48:58,830
now my equation is u prime.
808
00:48:58,830 --> 00:49:00,800
My first order
system is u prime,
809
00:49:00,800 --> 00:49:06,160
that'll be y double prime y
prime, the derivative of u,
810
00:49:06,160 --> 00:49:10,940
okay, now the differential
equation is telling me that y
811
00:49:10,940 --> 00:49:14,430
double prime is m- so
I'm just looking for --
812
00:49:14,430 --> 00:49:17,160
what's this matrix?
813
00:49:17,160 --> 00:49:19,410
y prime y.
814
00:49:19,410 --> 00:49:23,220
I'm looking for the matrix A.
815
00:49:23,220 --> 00:49:28,460
What's the matrix in case I have
a single first order equation
816
00:49:28,460 --> 00:49:31,140
and I want to make it
into a two by two system?
817
00:49:31,140 --> 00:49:32,270
Okay, simple.
818
00:49:32,270 --> 00:49:35,920
The first row of the matrix
is given by the equation.
819
00:49:35,920 --> 00:49:43,800
So y''-by'-ky -- no problem.
820
00:49:43,800 --> 00:49:47,240
And what's the second
row on the matrix?
821
00:49:47,240 --> 00:49:48,660
Then we're done.
822
00:49:48,660 --> 00:49:50,710
The second row of the
matrix I want just
823
00:49:50,710 --> 00:49:54,490
to be the trivial equation
y prime equals y prime,
824
00:49:54,490 --> 00:49:56,340
so I need a one
and a zero there.
825
00:49:59,240 --> 00:50:03,950
So matrixes like these,
the gen- the general case,
826
00:50:03,950 --> 00:50:09,050
if I had a five by five -- if
I had a fifth order equation
827
00:50:09,050 --> 00:50:11,590
and I wanted a five
by five matrix,
828
00:50:11,590 --> 00:50:15,850
I would see the coefficients of
the equation up there and then
829
00:50:15,850 --> 00:50:21,260
my four trivial equations
would put ones here.
830
00:50:21,260 --> 00:50:27,110
This is the kind of matrix
that goes from a fifth order
831
00:50:27,110 --> 00:50:32,060
to a five by five first order.
832
00:50:35,140 --> 00:50:40,010
So the -- and the eigenvalues
will come out in a natural way
833
00:50:40,010 --> 00:50:41,350
connected to the differential
834
00:50:41,350 --> 00:50:41,970
equation.
835
00:50:41,970 --> 00:50:45,840
Okay, that's
differential equations.
836
00:50:45,840 --> 00:50:49,890
The -- a parallel lecture
compared to powers of a matrix
837
00:50:49,890 --> 00:50:52,060
we can now do exponentials.
838
00:50:52,060 --> 00:50:53,610
Thanks.