1
00:00:02,632 --> 00:00:04,090
GILBERT STRANG:
This is a good time
2
00:00:04,090 --> 00:00:08,670
to do two by two matrices,
their eigenvalues,
3
00:00:08,670 --> 00:00:09,715
and their stability.
4
00:00:12,360 --> 00:00:16,470
Two by two eigenvalues
are the easiest
5
00:00:16,470 --> 00:00:19,390
to do, easiest to understand.
6
00:00:19,390 --> 00:00:23,930
Good to separate out the two
by two case from the later n
7
00:00:23,930 --> 00:00:25,980
by n eigenvalue problem.
8
00:00:28,610 --> 00:00:37,070
And of course, let me remember
the basic dogma of eigenvalues
9
00:00:37,070 --> 00:00:38,360
and eigenvectors.
10
00:00:38,360 --> 00:00:43,440
We're looking for a vector,
x, and a number, lambda,
11
00:00:43,440 --> 00:00:47,840
the eigenvalue, so
that Ax is lambda x.
12
00:00:47,840 --> 00:00:52,490
In other words, when
I multiply by A,
13
00:00:52,490 --> 00:00:56,820
that special vector x
does not change direction.
14
00:00:56,820 --> 00:01:01,540
It just changes length
by a factor lambda,
15
00:01:01,540 --> 00:01:03,290
which could be positive.
16
00:01:03,290 --> 00:01:05,040
It could be zero.
17
00:01:05,040 --> 00:01:06,500
Could be negative.
18
00:01:06,500 --> 00:01:08,240
Could be complex number.
19
00:01:08,240 --> 00:01:09,820
It's a number, though.
20
00:01:09,820 --> 00:01:15,150
So that's the key equation.
21
00:01:15,150 --> 00:01:17,990
Let me go toward its solution.
22
00:01:17,990 --> 00:01:22,670
So I want to move that
onto the left hand side.
23
00:01:22,670 --> 00:01:26,230
So I just write the
same equation this way.
24
00:01:26,230 --> 00:01:33,160
And now I see that this matrix
times the vector gives me 0.
25
00:01:33,160 --> 00:01:36,580
Now, when is that possible?
26
00:01:36,580 --> 00:01:39,780
That matrix can't be invertible.
27
00:01:39,780 --> 00:01:44,000
If it was invertible, the only
solution would be x equals 0.
28
00:01:44,000 --> 00:01:45,340
No good.
29
00:01:45,340 --> 00:01:50,370
So this matrix must be singular.
30
00:01:50,370 --> 00:01:52,370
It's determined it must be 0.
31
00:01:52,370 --> 00:01:56,740
And now we have an equation
for the eigenvalue lambda.
32
00:01:56,740 --> 00:02:00,830
So lambda is how much
we shift the matrix
33
00:02:00,830 --> 00:02:03,380
to make the determinant 0.
34
00:02:03,380 --> 00:02:07,220
We shift by lambda
times the identity
35
00:02:07,220 --> 00:02:10,789
to subtract that
from the diagonal.
36
00:02:10,789 --> 00:02:15,950
So can I begin with very
easy two by two matrix,
37
00:02:15,950 --> 00:02:21,450
the kind that we met first,
called a companion matrix.
38
00:02:21,450 --> 00:02:27,980
So we met this matrix when we
had a second order equation.
39
00:02:27,980 --> 00:02:34,790
So I started with the equation y
double prime plus By prime plus
40
00:02:34,790 --> 00:02:38,750
Cy equals, say, 0.
41
00:02:38,750 --> 00:02:43,150
So I started with one
second order equation.
42
00:02:43,150 --> 00:02:48,310
And then I introduced y
prime as a second unknown.
43
00:02:48,310 --> 00:02:52,250
So now I have a vector
unknown, y and y prime.
44
00:02:52,250 --> 00:02:56,610
And then, when I wrote
the equation down--
45
00:02:56,610 --> 00:03:01,930
I won't repeat that-- it led
us to a two by two matrix.
46
00:03:01,930 --> 00:03:05,910
Two equations for two
unknowns, y and y prime.
47
00:03:05,910 --> 00:03:10,410
So there is a two by two matrix
that we're interested in.
48
00:03:10,410 --> 00:03:14,880
But we really are going to be
interested in all two by twos.
49
00:03:14,880 --> 00:03:20,330
So let me take that to be my
matrix A, my companion matrix.
50
00:03:20,330 --> 00:03:23,290
So I just want to
go through the steps
51
00:03:23,290 --> 00:03:25,550
of finding its eigenvalues.
52
00:03:25,550 --> 00:03:29,760
What are the eigenvalues
of that matrix?
53
00:03:29,760 --> 00:03:33,940
We just take the
matrix, subtract lambda
54
00:03:33,940 --> 00:03:39,490
from the diagonal, and
take the determinant.
55
00:03:39,490 --> 00:03:43,140
And when I take the determinant
of a two by two matrix,
56
00:03:43,140 --> 00:03:46,200
it's just that
times that, which is
57
00:03:46,200 --> 00:03:49,940
minus lambda times minus
lambda is lambda squared.
58
00:03:49,940 --> 00:03:52,300
This gives me a B lambda.
59
00:03:52,300 --> 00:03:55,180
And the other part
of the determinant
60
00:03:55,180 --> 00:03:59,380
is this product, minus C. But
it comes with a minus sign,
61
00:03:59,380 --> 00:04:04,920
so it's plus C. So there's my
equation for the eigenvalues
62
00:04:04,920 --> 00:04:09,490
of a companion matrix.
63
00:04:09,490 --> 00:04:14,340
And of course you see that's
exactly the same equation
64
00:04:14,340 --> 00:04:18,750
that we had for the exponent s.
65
00:04:18,750 --> 00:04:28,680
So lambda for the matrix case
is the same as s, s1 and s2
66
00:04:28,680 --> 00:04:35,300
for the single second
order equation.
67
00:04:35,300 --> 00:04:39,760
So this equation has
solutions e to the st
68
00:04:39,760 --> 00:04:46,010
when the matrix has the
eigenvalues lambda equal s.
69
00:04:46,010 --> 00:04:49,870
Those same s1 and s2.
70
00:04:49,870 --> 00:04:57,260
But now I move on to a
general two by two matrix.
71
00:04:57,260 --> 00:04:59,630
What are its eigenvalues?
72
00:04:59,630 --> 00:05:03,470
What does that equation looks
like for its two eigenvalues?
73
00:05:03,470 --> 00:05:07,280
So this will be a
special case of this.
74
00:05:07,280 --> 00:05:12,070
Here, I have a general
matrix, a, b, c, d.
75
00:05:12,070 --> 00:05:14,880
I've subtracted lambda
from the diagonal.
76
00:05:14,880 --> 00:05:16,750
I'm taking the determinant.
77
00:05:16,750 --> 00:05:19,950
That'll give me the
two eigenvalues.
78
00:05:19,950 --> 00:05:22,170
Let's do it.
79
00:05:22,170 --> 00:05:26,140
Minus lambda times minus
lambda is lambda squared.
80
00:05:26,140 --> 00:05:30,190
Then I have a minus lambda
d and a minus lambda a.
81
00:05:30,190 --> 00:05:34,900
So I have an a plus a d lambda.
82
00:05:34,900 --> 00:05:37,880
And then I have the part
that doesn't involve lambda.
83
00:05:37,880 --> 00:05:40,530
The part that doesn't
involve lambda
84
00:05:40,530 --> 00:05:44,450
is just the determinant
of a, b, c, d.
85
00:05:44,450 --> 00:05:47,660
It's just the ad
and the minus bc.
86
00:05:47,660 --> 00:05:53,160
So there's an ad and a
minus bc, and all that is 0.
87
00:05:56,700 --> 00:05:59,600
It's a quadratic
equation, second degree.
88
00:05:59,600 --> 00:06:02,750
A two by two matrix
has two eigenvalues,
89
00:06:02,750 --> 00:06:05,990
the two roots of that equation.
90
00:06:05,990 --> 00:06:10,120
I just want to understand
more and more and more
91
00:06:10,120 --> 00:06:13,410
about the connection of
the roots, lambda 1 lambda
92
00:06:13,410 --> 00:06:17,640
2, to the matrix a, b, c, d.
93
00:06:17,640 --> 00:06:22,610
If I know the two by two matrix,
this tells me the eigenvalues.
94
00:06:22,610 --> 00:06:28,710
So this will, being a quadratic
equation, have two roots.
95
00:06:31,510 --> 00:06:36,290
So if I factor this, this will
factor into lambda minus lambda
96
00:06:36,290 --> 00:06:40,980
1 times lambda minus lambda 2.
97
00:06:40,980 --> 00:06:43,710
And of course, if
the numbers are nice,
98
00:06:43,710 --> 00:06:48,270
then I can see what
lambda 1 and lambda 2 are.
99
00:06:48,270 --> 00:06:53,080
In that case, I find
the eigenvalues.
100
00:06:53,080 --> 00:06:57,210
If the numbers are not nice,
then lambda 1 and lambda 2
101
00:06:57,210 --> 00:07:01,460
come from the quadratic formula,
the minus b plus or minus
102
00:07:01,460 --> 00:07:05,350
square root of b
squared minus 4ac.
103
00:07:05,350 --> 00:07:09,590
The quadratic formula will solve
this equation, will tell me
104
00:07:09,590 --> 00:07:11,130
these two numbers.
105
00:07:11,130 --> 00:07:17,420
And if I multiply it out this
way, I see lambda squared.
106
00:07:17,420 --> 00:07:24,450
I see minus lambda times
lambda 1 and lambda 2.
107
00:07:27,180 --> 00:07:34,710
And then I see plus lambda
1 times lambda 2 equals 0.
108
00:07:37,270 --> 00:07:41,360
Here, I've written the
equation for the two lambdas.
109
00:07:41,360 --> 00:07:45,740
Here, I've written the equation
when I know the two lambdas.
110
00:07:45,740 --> 00:07:47,200
Why did I do this?
111
00:07:47,200 --> 00:07:50,710
I want to match this
with this and see
112
00:07:50,710 --> 00:07:57,720
that this number, whatever it
is, is the same as that number.
113
00:07:57,720 --> 00:08:01,130
They show up there, the
coefficient of minus lambda.
114
00:08:01,130 --> 00:08:09,380
So that's the first step,
that lambda 1 plus lambda 2
115
00:08:09,380 --> 00:08:12,020
is the same as a plus d.
116
00:08:15,400 --> 00:08:18,890
Just matching those
two equations.
117
00:08:18,890 --> 00:08:23,890
This is just like a general
fact about a quadratic equation.
118
00:08:23,890 --> 00:08:30,370
The sum of the roots is the
minus coefficient of lambda.
119
00:08:30,370 --> 00:08:35,679
And then the constant
term is the constant term.
120
00:08:35,679 --> 00:08:42,183
So lambda 1 times
lambda 2 is ad minus bc.
121
00:08:46,880 --> 00:08:53,850
These are facts about a two
by two matrix, a, b, c, d.
122
00:08:53,850 --> 00:08:55,650
The sum of the eigenvalues.
123
00:08:55,650 --> 00:08:57,670
So this is the sum
of the eigenvalues--
124
00:08:57,670 --> 00:09:01,790
so I'll put s-u-m to
indicate that I'm looking
125
00:09:01,790 --> 00:09:06,470
at the sum-- is that a plus d.
126
00:09:06,470 --> 00:09:09,420
A plus d are the
numbers on the diagonal.
127
00:09:09,420 --> 00:09:11,350
So that's a little special.
128
00:09:11,350 --> 00:09:13,950
When I add the
diagonal numbers, I
129
00:09:13,950 --> 00:09:19,590
get something called
the trace of the matrix.
130
00:09:23,210 --> 00:09:25,580
I'm introducing a word, trace.
131
00:09:25,580 --> 00:09:28,750
Trace is the add up
down the diagonal.
132
00:09:28,750 --> 00:09:30,240
And that matches a plus d.
133
00:09:30,740 --> 00:09:38,460
And this one is the product
of the eigenvalues lambda
134
00:09:38,460 --> 00:09:40,180
1 times lambda 2.
135
00:09:40,180 --> 00:09:41,650
So that's the product.
136
00:09:41,650 --> 00:09:46,110
And that's equal to
the determinant of a.
137
00:09:49,490 --> 00:09:53,890
I'm just making all the
neat connections that
138
00:09:53,890 --> 00:09:57,700
are special for a two by two.
139
00:09:57,700 --> 00:10:00,920
So that if I write
down some matrices,
140
00:10:00,920 --> 00:10:02,780
we could look at
them immediately.
141
00:10:02,780 --> 00:10:05,920
Let me write down a matrix.
142
00:10:05,920 --> 00:10:07,630
Suppose I write
down that matrix.
143
00:10:14,140 --> 00:10:25,320
Oh, let me make them
0, 1-- well, 0, 4-- ah,
144
00:10:25,320 --> 00:10:27,170
let me improve this a little.
145
00:10:27,170 --> 00:10:30,481
2, 4, 4, 9.
146
00:10:30,481 --> 00:10:32,351
2, 4, 4, 2 would be even easier.
147
00:10:32,351 --> 00:10:32,850
Sorry.
148
00:10:36,420 --> 00:10:38,810
I look at that matrix.
149
00:10:38,810 --> 00:10:42,730
I see immediately the two
eigenvalues of that matrix
150
00:10:42,730 --> 00:10:44,646
add to 4.
151
00:10:44,646 --> 00:10:46,200
2 plus 2 is 4.
152
00:10:46,200 --> 00:10:48,400
I took the trace.
153
00:10:48,400 --> 00:10:51,440
The two eigenvalues of
that matrix multiply
154
00:10:51,440 --> 00:10:58,110
to the determinant, which is 2
times 2 is 4 minus 16 minus 12.
155
00:10:58,110 --> 00:11:03,350
So the sum here for
that matrix would be 4.
156
00:11:03,350 --> 00:11:06,860
The determinant of that
matrix would be 4 minus 16
157
00:11:06,860 --> 00:11:08,760
is minus 12.
158
00:11:08,760 --> 00:11:15,980
And maybe I can come up with the
two numbers that have add to 4
159
00:11:15,980 --> 00:11:17,750
and multiply to minus 12.
160
00:11:17,750 --> 00:11:22,285
I think, actually, that
they are six and minus 2.
161
00:11:22,285 --> 00:11:28,370
I think that the eigenvalues
here are 6 and minus 2
162
00:11:28,370 --> 00:11:31,500
because those add
up to 4, the trace,
163
00:11:31,500 --> 00:11:35,630
and they multiply 6 times
minus 2 is minus 12.
164
00:11:35,630 --> 00:11:38,550
That's the determinant.
165
00:11:38,550 --> 00:11:41,790
Two by two matrices,
you have a good chance
166
00:11:41,790 --> 00:11:44,590
at seeing exactly what happens.
167
00:11:44,590 --> 00:11:53,630
Now, my interest today for
this video is to use all this,
168
00:11:53,630 --> 00:11:58,220
use the eigenvalues,
to decide stability.
169
00:11:58,220 --> 00:12:02,540
Stability means that the
differential equation
170
00:12:02,540 --> 00:12:05,700
has solutions that go to 0.
171
00:12:05,700 --> 00:12:11,110
And we remember
the solutions are
172
00:12:11,110 --> 00:12:16,330
e to the st, which is the
same as e to the lambda t.
173
00:12:16,330 --> 00:12:20,520
The s and the lambda both
come from that same equation
174
00:12:20,520 --> 00:12:27,770
in the case of a second order
equation reduced to a companion
175
00:12:27,770 --> 00:12:28,890
matrix.
176
00:12:28,890 --> 00:12:37,560
So I'm interested in when
are the eigenvalues negative.
177
00:12:37,560 --> 00:12:39,740
When are the
eigenvalues negative?
178
00:12:39,740 --> 00:12:42,500
Or if they're
complex numbers, when
179
00:12:42,500 --> 00:12:44,720
are their real parts negative.
180
00:12:44,720 --> 00:12:51,400
So can we remember trace, the
sum, product, the determinant.
181
00:12:51,400 --> 00:12:54,820
And answer the
stability questions.
182
00:12:54,820 --> 00:12:55,995
So I'm ready for stability.
183
00:12:59,620 --> 00:13:04,090
So stability means
either lambda 1 negative
184
00:13:04,090 --> 00:13:07,710
and lambda 2 negative.
185
00:13:07,710 --> 00:13:09,145
This is in the real case.
186
00:13:11,840 --> 00:13:21,270
Or in the complex case,
lambda equals some real part
187
00:13:21,270 --> 00:13:25,690
plus and minus some
imaginary part.
188
00:13:25,690 --> 00:13:29,190
Then we want the real
part to be negative.
189
00:13:29,190 --> 00:13:34,490
Real part of a lambda,
which is a, should be 0.
190
00:13:34,490 --> 00:13:36,460
So that's our requirement.
191
00:13:36,460 --> 00:13:38,750
If the eigenvalues
are complex, we
192
00:13:38,750 --> 00:13:42,010
get a pair of them
and the real part
193
00:13:42,010 --> 00:13:46,990
should be 0 so that e to the--
the point about this negative a
194
00:13:46,990 --> 00:13:52,020
is that e to the
at will go to 0.
195
00:13:52,020 --> 00:13:53,910
The point about these
negative lambdas
196
00:13:53,910 --> 00:13:58,650
is that e to the
lambda t will go to 0.
197
00:13:58,650 --> 00:14:01,630
This is stability.
198
00:14:01,630 --> 00:14:11,030
So my question is, what's the
test on the matrix that decides
199
00:14:11,030 --> 00:14:13,360
this about the eigenvalues?
200
00:14:15,980 --> 00:14:18,340
Can we look at
the matrix-- maybe
201
00:14:18,340 --> 00:14:21,580
we don't have to find
those eigenvalues.
202
00:14:21,580 --> 00:14:23,330
Maybe we can use the fact.
203
00:14:23,330 --> 00:14:27,340
Again, the fact is that
lambda 1 plus lambda 2
204
00:14:27,340 --> 00:14:36,790
is the trace and lambda 1 times
lambda 2 is the determinant.
205
00:14:36,790 --> 00:14:40,580
And we can read those
numbers off from the matrix.
206
00:14:40,580 --> 00:14:42,950
Then there's a
quadratic equation.
207
00:14:42,950 --> 00:14:47,690
But if we only want to
know information like
208
00:14:47,690 --> 00:14:50,560
are the eigenvalues negative?
209
00:14:50,560 --> 00:14:53,360
Are their real parts negative?
210
00:14:53,360 --> 00:14:57,490
We can get that information
from these numbers
211
00:14:57,490 --> 00:15:03,110
without going to
finding the eigenvalues
212
00:15:03,110 --> 00:15:05,110
from that quadratic equation.
213
00:15:05,110 --> 00:15:08,740
Wouldn't be that hard to do,
but we don't have to do it.
214
00:15:08,740 --> 00:15:13,430
So suppose we have two
negative eigenvalues.
215
00:15:13,430 --> 00:15:21,630
Then certainly, this would mean
the trace would be negative.
216
00:15:21,630 --> 00:15:25,110
Because the trace is the
sum of the eigenvalues.
217
00:15:25,110 --> 00:15:29,320
If those are both negative,
trace is negative.
218
00:15:29,320 --> 00:15:33,470
So we can check about the
trace just right away.
219
00:15:33,470 --> 00:15:35,530
What about the determinant?
220
00:15:35,530 --> 00:15:38,660
If that's negative
and that's negative,
221
00:15:38,660 --> 00:15:42,090
then multiplying those will
give a positive number.
222
00:15:42,090 --> 00:15:44,600
So the determinant
should be positive.
223
00:15:44,600 --> 00:15:47,320
So trace less than 0.
224
00:15:47,320 --> 00:15:50,680
Determinant greater than 0.
225
00:15:50,680 --> 00:15:53,550
That is the stability test.
226
00:15:53,550 --> 00:15:55,250
That's the stability test.
227
00:15:57,610 --> 00:15:58,110
Stable.
228
00:16:01,590 --> 00:16:06,230
The two by two matrix A, B,
C, D, if its trace is negative
229
00:16:06,230 --> 00:16:09,780
and its determinant is
positive, is stable.
230
00:16:09,780 --> 00:16:11,190
That's the test.
231
00:16:11,190 --> 00:16:16,690
And actually, it works also
if lambda comes out complex
232
00:16:16,690 --> 00:16:24,280
because lambda 1 plus lambda
2-- lambda 1 is a plus i omega.
233
00:16:24,280 --> 00:16:26,840
Lambda 2 is a minus omega.
234
00:16:26,840 --> 00:16:30,130
The sum is just 2a.
235
00:16:30,130 --> 00:16:32,500
And we want that to be negative.
236
00:16:32,500 --> 00:16:35,530
So again, trace negative.
237
00:16:35,530 --> 00:16:40,830
Trace negative even if the roots
are real or if they're complex.
238
00:16:40,830 --> 00:16:44,970
That still tells us that the
sum of the roots is negative
239
00:16:44,970 --> 00:16:47,450
and the determinant also works.
240
00:16:47,450 --> 00:16:53,980
If a plus i omega times a
minus i omega-- in this case,
241
00:16:53,980 --> 00:16:58,140
lambda 1 times lambda 2--
if I multiply those numbers,
242
00:16:58,140 --> 00:17:02,750
I get a squared
plus omega squared.
243
00:17:02,750 --> 00:17:04,260
With a plus.
244
00:17:04,260 --> 00:17:06,079
So that would be positive.
245
00:17:06,079 --> 00:17:09,089
And we're good.
246
00:17:09,089 --> 00:17:16,890
So my conclusion is this
is the test for stability.
247
00:17:16,890 --> 00:17:19,020
And I can apply it
to a few matrices.
248
00:17:19,020 --> 00:17:21,990
I wrote down a few matrices.
249
00:17:21,990 --> 00:17:26,880
Can I just look at that test--
can you look at that test--
250
00:17:26,880 --> 00:17:28,950
and just apply it to see.
251
00:17:31,980 --> 00:17:34,170
So here's an example.
252
00:17:34,170 --> 00:17:41,130
Say minus 2, minus 1, 3, and 4.
253
00:17:41,130 --> 00:17:42,740
Is that any good?
254
00:17:42,740 --> 00:17:45,470
The trace is minus 3.
255
00:17:45,470 --> 00:17:46,690
That's good.
256
00:17:46,690 --> 00:17:50,190
The determinant is
2 minus 12 minus 10.
257
00:17:50,190 --> 00:17:51,710
That's bad.
258
00:17:51,710 --> 00:17:52,850
That's bad.
259
00:17:52,850 --> 00:17:55,650
So that would be unstable.
260
00:17:58,460 --> 00:18:00,120
That has a negative determinant.
261
00:18:00,120 --> 00:18:01,570
Unstable.
262
00:18:01,570 --> 00:18:03,500
So I'll put an x through that.
263
00:18:03,500 --> 00:18:04,840
Unstable.
264
00:18:04,840 --> 00:18:07,210
Let me take a stable one.
265
00:18:07,210 --> 00:18:13,170
Stable one, I'm going to want
like minus 5, and 1, let's say.
266
00:18:13,170 --> 00:18:14,620
That's OK.
267
00:18:14,620 --> 00:18:16,350
The trace is negative.
268
00:18:16,350 --> 00:18:17,710
Minus 4.
269
00:18:17,710 --> 00:18:21,250
And now I want to make
the determinant positive.
270
00:18:21,250 --> 00:18:27,360
So maybe I better put
like 6 and minus 7.
271
00:18:27,360 --> 00:18:28,940
Just picking numbers.
272
00:18:28,940 --> 00:18:36,550
So now the determinant
is minus 5 plus 42.
273
00:18:36,550 --> 00:18:38,620
A big positive number.
274
00:18:38,620 --> 00:18:41,310
And the determinant
test is passed.
275
00:18:41,310 --> 00:18:43,330
So that is OK.
276
00:18:43,330 --> 00:18:44,490
That one would be stable.
277
00:18:47,420 --> 00:18:52,500
If this was my matrix
A, then the solutions
278
00:18:52,500 --> 00:19:01,450
to dy dt equal Ay, y prime equal
Ay is my differential equation.
279
00:19:01,450 --> 00:19:06,610
The two solutions which
would track the eigenvectors
280
00:19:06,610 --> 00:19:09,580
would have negative lambdas.
281
00:19:09,580 --> 00:19:13,120
Negative lambdas because
the trace is negative
282
00:19:13,120 --> 00:19:15,590
and the determinant is positive.
283
00:19:15,590 --> 00:19:18,510
Passes the stability
test and the solutions
284
00:19:18,510 --> 00:19:21,960
would go to minus infinity.
285
00:19:21,960 --> 00:19:23,900
That's two by twos.
286
00:19:23,900 --> 00:19:25,650
Thank you.