1
00:00:02,940 --> 00:00:08,330
GILBERT STRANG: So today begins
eigenvalues and eigenvectors.
2
00:00:08,330 --> 00:00:11,320
And the reason we
want those, need
3
00:00:11,320 --> 00:00:17,540
those is to solve systems
of linear equations.
4
00:00:17,540 --> 00:00:22,680
Systems meaning more than
one equation, n equations.
5
00:00:22,680 --> 00:00:27,090
n equal 2 in the examples here.
6
00:00:27,090 --> 00:00:32,000
So eigenvalue is a number,
eigenvector is a vector.
7
00:00:32,000 --> 00:00:35,600
They're both hiding
in the matrix.
8
00:00:35,600 --> 00:00:38,640
Once we find them,
we can use them.
9
00:00:38,640 --> 00:00:43,600
Let me show you the reason
eigenvalues were created,
10
00:00:43,600 --> 00:00:50,150
invented, discovered was solving
differential equations, which
11
00:00:50,150 --> 00:00:52,330
is our purpose.
12
00:00:52,330 --> 00:00:59,110
So why is now a vector-- so
this is a system of equations.
13
00:00:59,110 --> 00:01:01,590
I'll do an example in a minute.
14
00:01:01,590 --> 00:01:04,760
A is a matrix.
15
00:01:04,760 --> 00:01:09,460
So we have n equations,
n components of y.
16
00:01:09,460 --> 00:01:15,550
And A is an n by n
matrix, n rows, n columns.
17
00:01:15,550 --> 00:01:16,660
Good.
18
00:01:16,660 --> 00:01:20,600
And now I can tell you
right away where eigenvalues
19
00:01:20,600 --> 00:01:23,670
and eigenvectors pay off.
20
00:01:23,670 --> 00:01:26,160
They come into the solution.
21
00:01:26,160 --> 00:01:29,800
We look for solutions
of that kind.
22
00:01:29,800 --> 00:01:32,790
When we had one equation,
we looked for solutions
23
00:01:32,790 --> 00:01:37,890
just e to the st, and
we found that number s.
24
00:01:37,890 --> 00:01:41,540
Now we have e to the
lambda t-- we changed s
25
00:01:41,540 --> 00:01:46,120
to lambda, no problem--
but multiplied by a vector
26
00:01:46,120 --> 00:01:49,610
because our unknown is a vector.
27
00:01:49,610 --> 00:01:52,830
This is a vector, but that
does not depend on time.
28
00:01:52,830 --> 00:01:54,510
That's the beauty of it.
29
00:01:54,510 --> 00:01:59,330
All the time dependence is in
the exponential, as always.
30
00:01:59,330 --> 00:02:05,870
And x is just multiples of that
exponential, as you'll see.
31
00:02:05,870 --> 00:02:08,550
So I look for
solutions like that.
32
00:02:08,550 --> 00:02:12,970
I plug that into the
differential equation
33
00:02:12,970 --> 00:02:14,510
and what happens?
34
00:02:14,510 --> 00:02:16,140
So here's my equation.
35
00:02:16,140 --> 00:02:21,110
I'm plugging in what e to
the lambda tx, that's y.
36
00:02:21,110 --> 00:02:23,520
That's A times y there.
37
00:02:23,520 --> 00:02:27,480
Now, the derivative of
y, the time derivative,
38
00:02:27,480 --> 00:02:30,750
brings down a lambda.
39
00:02:30,750 --> 00:02:33,520
To get the derivative
I include the lambda.
40
00:02:33,520 --> 00:02:37,850
So do you see that
substituting into the equation
41
00:02:37,850 --> 00:02:43,360
with this nice notation is
just this has to be true.
42
00:02:43,360 --> 00:02:47,050
My equation changed
to that form.
43
00:02:47,050 --> 00:02:51,430
OK Now I cancel either
the lambda t, just the way
44
00:02:51,430 --> 00:02:54,980
I was always
canceling e to the st.
45
00:02:54,980 --> 00:02:58,700
So I cancel e to the lambda
t because it's never zero.
46
00:02:58,700 --> 00:03:04,080
And I have the big
equation, Ax, the matrix
47
00:03:04,080 --> 00:03:07,920
times my eigenvector,
is equal to lambda
48
00:03:07,920 --> 00:03:13,680
x-- the number, the eigenvalue,
times the eigenvector.
49
00:03:13,680 --> 00:03:16,140
Not linear, notice.
50
00:03:16,140 --> 00:03:18,210
Two unknowns here
that are multiplied.
51
00:03:18,210 --> 00:03:21,820
A number, lambda,
times a vector, x.
52
00:03:21,820 --> 00:03:23,960
So what am I looking for?
53
00:03:23,960 --> 00:03:29,440
I'm looking for vectors x,
the eigenvectors, so that
54
00:03:29,440 --> 00:03:36,700
multiplying by A-- multiplying A
times x gives a number times x.
55
00:03:36,700 --> 00:03:41,870
It's in the same direction as
x just the length is changed.
56
00:03:41,870 --> 00:03:46,590
Well, if lambda was 1,
I would have Ax equal x.
57
00:03:46,590 --> 00:03:47,850
That's allowed.
58
00:03:47,850 --> 00:03:51,400
If lambda is 0, I
would have Ax equals 0.
59
00:03:51,400 --> 00:03:52,810
That's all right.
60
00:03:52,810 --> 00:03:54,640
I don't want x to be 0.
61
00:03:54,640 --> 00:03:57,085
That's useless.
62
00:03:57,085 --> 00:04:01,780
That's no help to know
that 0 is a solution.
63
00:04:01,780 --> 00:04:04,070
So x should be not 0.
64
00:04:04,070 --> 00:04:06,990
Lambda can be any number.
65
00:04:06,990 --> 00:04:09,880
It can be real, it
could be complex number,
66
00:04:09,880 --> 00:04:11,120
as you will see.
67
00:04:11,120 --> 00:04:14,860
Even if the matrix is real,
lambda could be complex.
68
00:04:14,860 --> 00:04:17,190
Anyway, Ax equal lambda x.
69
00:04:17,190 --> 00:04:19,029
That's the big equation.
70
00:04:19,029 --> 00:04:21,690
It got a box around it.
71
00:04:21,690 --> 00:04:26,820
So now I'm ready
to do an example.
72
00:04:26,820 --> 00:04:29,460
And in this example,
first of all,
73
00:04:29,460 --> 00:04:33,670
I'm going to spot the
eigenvalues and eigenvectors
74
00:04:33,670 --> 00:04:39,090
without a system, just go
for it in the 2 by 2 case.
75
00:04:39,090 --> 00:04:43,360
So I'll give a 2 by 2 matrix
A. We'll find the lambdas
76
00:04:43,360 --> 00:04:48,690
and the x's, and then we'll
have the solution to the system
77
00:04:48,690 --> 00:04:50,180
of differential equations.
78
00:04:50,180 --> 00:04:53,050
Good.
79
00:04:53,050 --> 00:04:55,420
There's the system.
80
00:04:55,420 --> 00:05:00,620
There's the first equation for
y1-- prime meaning derivative,
81
00:05:00,620 --> 00:05:06,220
d by dt, time derivative-- is
linear, a constant coefficient.
82
00:05:06,220 --> 00:05:10,280
Second one, linear, constant
coefficient, 3 and 3.
83
00:05:10,280 --> 00:05:15,540
Those numbers, 5, 1, 3,
3, go into the matrix.
84
00:05:15,540 --> 00:05:21,970
Then that problem is exactly y
prime, the vector, derivative
85
00:05:21,970 --> 00:05:25,890
of the vector, equal A times y.
86
00:05:25,890 --> 00:05:27,380
That's my problem.
87
00:05:27,380 --> 00:05:32,800
Now eigenvalues and
eigenvectors will solve it.
88
00:05:32,800 --> 00:05:36,340
So I just look at that matrix.
89
00:05:36,340 --> 00:05:37,870
Matrix question.
90
00:05:37,870 --> 00:05:41,360
What are the eigenvalues,
what are the eigenvectors
91
00:05:41,360 --> 00:05:43,270
of that matrix?
92
00:05:43,270 --> 00:05:52,182
And remember, I want
Ax equals lambda x.
93
00:05:52,182 --> 00:05:55,680
I've spotted the
first eigenvector.
94
00:05:55,680 --> 00:05:58,370
1, 1.
95
00:05:58,370 --> 00:06:00,750
We could just
check does it work.
96
00:06:00,750 --> 00:06:04,340
If I multiply A by
that eigenvector,
97
00:06:04,340 --> 00:06:08,350
1, 1, do you see what
happens when I multiply by 1?
98
00:06:08,350 --> 00:06:10,380
That gives me a 6.
99
00:06:10,380 --> 00:06:12,700
That gives me a 6.
100
00:06:12,700 --> 00:06:17,210
So A times that vector is 6, 6.
101
00:06:17,210 --> 00:06:20,870
And that is 6 times 1, 1.
102
00:06:20,870 --> 00:06:22,310
So there you go.
103
00:06:22,310 --> 00:06:24,610
Found the first eigenvalue.
104
00:06:24,610 --> 00:06:29,050
If I multiply A by
x, I get 6 by x.
105
00:06:29,050 --> 00:06:31,420
I get the vector 6, 6.
106
00:06:31,420 --> 00:06:33,440
Now, the second one.
107
00:06:33,440 --> 00:06:39,170
Again, I've worked in advance,
produced this eigenvector,
108
00:06:39,170 --> 00:06:42,220
and I think it's 1 minus 3.
109
00:06:42,220 --> 00:06:46,430
So let's multiply by A.
Try the second eigenvector.
110
00:06:46,430 --> 00:06:51,010
I should call this first
one maybe x1 and lambda 1.
111
00:06:51,010 --> 00:06:54,070
And I should call this
one x2 and lambda 2.
112
00:06:54,070 --> 00:06:58,400
And we can find out what
lambda 2 is, once I find
113
00:06:58,400 --> 00:07:00,160
the eigenvectors of course.
114
00:07:00,160 --> 00:07:06,060
I just do A times x to recognize
the lambda, the eigenvalue.
115
00:07:06,060 --> 00:07:11,580
So 5, 1 times this
is 5 minus 3 is a 2.
116
00:07:11,580 --> 00:07:14,250
It's a 2.
117
00:07:14,250 --> 00:07:17,680
So here I got a 2.
118
00:07:17,680 --> 00:07:25,020
And from 3, 3 it's 3
minus 9 is minus 6.
119
00:07:25,020 --> 00:07:28,410
That's what I got for Ax.
120
00:07:28,410 --> 00:07:31,020
There was the x.
121
00:07:31,020 --> 00:07:34,920
When I did the multiplication,
Ax came out to be 2 minus 6.
122
00:07:34,920 --> 00:07:36,360
Good.
123
00:07:36,360 --> 00:07:41,380
That output is two
times the input.
124
00:07:41,380 --> 00:07:44,050
The eigenvalue is 2.
125
00:07:44,050 --> 00:07:45,370
Right?
126
00:07:45,370 --> 00:07:49,560
I'm looking for inputs,
the eigenvector,
127
00:07:49,560 --> 00:07:54,420
so that the output is a
number times that eigenvector,
128
00:07:54,420 --> 00:07:57,730
and that number is
lambda, the eigenvalue.
129
00:07:57,730 --> 00:08:01,510
So I've now found the two.
130
00:08:01,510 --> 00:08:04,730
And I expect two
for a 2 by 2 matrix.
131
00:08:04,730 --> 00:08:08,820
You will soon see why I
expect two eigenvalues,
132
00:08:08,820 --> 00:08:12,380
and each eigenvalue should
have an eigenvector.
133
00:08:12,380 --> 00:08:17,050
So here they are
for this matrix.
134
00:08:17,050 --> 00:08:19,730
So I've got the answers now.
135
00:08:19,730 --> 00:08:28,400
y of t, which stands
for y1 and y2 of t.
136
00:08:30,960 --> 00:08:36,280
Those are-- it's e
to the lambda tx.
137
00:08:36,280 --> 00:08:39,470
Remember, that's the picture
that we're looking for.
138
00:08:39,470 --> 00:08:47,770
So the first one is e to the
6t times x, which is 1, 1.
139
00:08:47,770 --> 00:08:52,490
If I put that into the equation,
it will solve the equation.
140
00:08:52,490 --> 00:08:55,502
Also, I have another one.
141
00:08:55,502 --> 00:09:00,330
e to the lambda 2 was 2t.
142
00:09:00,330 --> 00:09:06,330
e to the lambda t times
its eigenvector, 1 minus 3.
143
00:09:06,330 --> 00:09:08,520
That's a solution also.
144
00:09:08,520 --> 00:09:10,490
One solution, another solution.
145
00:09:10,490 --> 00:09:13,350
And what do I do with
linear equations?
146
00:09:13,350 --> 00:09:16,040
I take combinations.
147
00:09:16,040 --> 00:09:21,090
Any number c1 of that,
plus any number c2 of that
148
00:09:21,090 --> 00:09:23,370
is still a solution.
149
00:09:23,370 --> 00:09:28,890
That's superposition, adding
solutions to linear equations.
150
00:09:28,890 --> 00:09:31,500
These are null equations.
151
00:09:31,500 --> 00:09:35,130
There's no force term
in these equations.
152
00:09:35,130 --> 00:09:36,740
I'm not dealing
with a force term.
153
00:09:36,740 --> 00:09:40,800
I'm looking for the null
solutions, the solutions
154
00:09:40,800 --> 00:09:43,320
of the equations themselves.
155
00:09:43,320 --> 00:09:50,510
And there I have two solutions,
two coefficients to choose.
156
00:09:50,510 --> 00:09:52,290
How do I choose them?
157
00:09:52,290 --> 00:09:57,130
Of course, I match the initial
condition, so at t equals 0.
158
00:09:59,990 --> 00:10:01,700
At t equals 0.
159
00:10:01,700 --> 00:10:06,710
At t equals 0, I
would have y of 0.
160
00:10:10,160 --> 00:10:16,350
That's my given initial
condition, my y1 and y2.
161
00:10:16,350 --> 00:10:20,480
So I'm setting t equals 0,
so that's one of course.
162
00:10:20,480 --> 00:10:22,530
When t is 0, that's one.
163
00:10:22,530 --> 00:10:27,240
So I just have c1 times 1, 1.
164
00:10:27,240 --> 00:10:38,980
And c2-- that's one again at
t equals o-- times 1 minus 3.
165
00:10:38,980 --> 00:10:44,230
That's what
determines c1 and c2.
166
00:10:44,230 --> 00:10:47,800
c1 and c2 come from
the initial conditions
167
00:10:47,800 --> 00:10:51,280
just the way they always did.
168
00:10:51,280 --> 00:10:56,600
So I'm solving two first order
linear constant coefficient
169
00:10:56,600 --> 00:11:01,760
equations, homogeneous,
meaning no force term.
170
00:11:05,360 --> 00:11:08,850
So I get a null solution
with constants to choose
171
00:11:08,850 --> 00:11:13,370
and, as always, those
constants come from matching
172
00:11:13,370 --> 00:11:15,690
the initial conditions.
173
00:11:15,690 --> 00:11:19,150
So the initial condition
here is a vector.
174
00:11:21,780 --> 00:11:27,580
So if, for example, y
of 0 was 2 minus 2, then
175
00:11:27,580 --> 00:11:32,500
I would want one of
those and one of those.
176
00:11:32,500 --> 00:11:34,740
OK.
177
00:11:34,740 --> 00:11:37,650
I've used eigenvalues
and eigenvectors
178
00:11:37,650 --> 00:11:43,530
to solve a linear system, their
first and primary purpose.
179
00:11:43,530 --> 00:11:44,610
OK.
180
00:11:44,610 --> 00:11:49,240
But how do I find those
eigenvalues and eigenvectors?
181
00:11:49,240 --> 00:11:51,250
What about other properties?
182
00:11:51,250 --> 00:11:54,220
What's going on with
eigenvalues and eigenvectors?
183
00:11:54,220 --> 00:11:58,340
May I begin on this just
a couple more minutes
184
00:11:58,340 --> 00:12:00,970
about eigenvalues
and eigenvectors?
185
00:12:00,970 --> 00:12:07,390
Basic facts and then I'll come
next video of how to find them.
186
00:12:07,390 --> 00:12:10,000
OK, basic facts.
187
00:12:10,000 --> 00:12:10,910
Basic facts.
188
00:12:10,910 --> 00:12:21,570
So start from Ax
equals lambda x.
189
00:12:21,570 --> 00:12:25,350
Let's suppose we found those.
190
00:12:25,350 --> 00:12:28,620
Could you tell me the
eigenvalues and eigenvectors
191
00:12:28,620 --> 00:12:29,480
of A squared?
192
00:12:35,120 --> 00:12:38,690
I would like to know what the
eigenvalues and eigenvectors
193
00:12:38,690 --> 00:12:39,860
of A squared are.
194
00:12:39,860 --> 00:12:42,190
Are they connected with these?
195
00:12:42,190 --> 00:12:48,240
So suppose I know the x and
I know the lambda for A. What
196
00:12:48,240 --> 00:12:50,200
about for A squared?
197
00:12:50,200 --> 00:12:53,800
Well, the good thing is
that the eigenvectors
198
00:12:53,800 --> 00:12:56,210
are the same for A squared.
199
00:12:56,210 --> 00:12:58,570
So let me show you.
200
00:12:58,570 --> 00:13:06,650
I say that same x, so this
is the same x, same vector,
201
00:13:06,650 --> 00:13:07,980
same eigenvector.
202
00:13:07,980 --> 00:13:10,220
The eigenvalue would be
different, of course,
203
00:13:10,220 --> 00:13:15,390
for A squared, but the
eigenvector is the same.
204
00:13:15,390 --> 00:13:18,050
And let's see what
happens for A squared.
205
00:13:18,050 --> 00:13:23,010
So that's A times Ax, right?
206
00:13:23,010 --> 00:13:26,090
One A, another Ax.
207
00:13:26,090 --> 00:13:30,132
But Ax is lambda x.
208
00:13:30,132 --> 00:13:31,090
Are you good with that?
209
00:13:31,090 --> 00:13:32,990
That's just A times Ax.
210
00:13:32,990 --> 00:13:34,970
So that's OK.
211
00:13:34,970 --> 00:13:37,250
Now lambda is a number.
212
00:13:37,250 --> 00:13:42,070
I like to bring it out
front where I can see it.
213
00:13:42,070 --> 00:13:44,440
So I didn't do anything there.
214
00:13:44,440 --> 00:13:47,080
This number lambda was
multiplying everything
215
00:13:47,080 --> 00:13:48,770
so I put it in front.
216
00:13:48,770 --> 00:13:49,740
Now Ax.
217
00:13:49,740 --> 00:13:52,020
I have, again, the Ax.
218
00:13:52,020 --> 00:13:54,830
That's, again, the
lambda x because I'm
219
00:13:54,830 --> 00:13:56,480
looking at the same x.
220
00:13:56,480 --> 00:13:59,220
Same x, so I get
the same lambda.
221
00:13:59,220 --> 00:14:02,120
So that's a lambda
x, another lambda.
222
00:14:02,120 --> 00:14:05,150
I have lambda squared x.
223
00:14:05,150 --> 00:14:06,910
That's what I wanted.
224
00:14:06,910 --> 00:14:11,040
A squared x is lambda squared x.
225
00:14:11,040 --> 00:14:12,370
Conclusion.
226
00:14:12,370 --> 00:14:22,030
The eigenvectors stay the same,
lambda goes to lambda squared.
227
00:14:22,030 --> 00:14:24,450
The eigenvalues are squared.
228
00:14:24,450 --> 00:14:31,175
So if I had my example again--
oh, let me find that matrix.
229
00:14:36,860 --> 00:14:40,590
Suppose I had that
same matrix and I
230
00:14:40,590 --> 00:14:43,740
was interested in
A squared, then
231
00:14:43,740 --> 00:14:50,480
the eigenvalues would be
36 and 4, the squares.
232
00:14:50,480 --> 00:14:53,300
I suppose I'm looking at
the n-th power of a matrix.
233
00:14:53,300 --> 00:14:55,780
You may say why look
at the n-th power?
234
00:14:55,780 --> 00:14:57,580
But there are many
examples to look
235
00:14:57,580 --> 00:15:01,340
at the n-th power of a
matrix, the thousandth power.
236
00:15:01,340 --> 00:15:04,670
So let's just write
down the conclusion.
237
00:15:04,670 --> 00:15:13,710
Same reasoning, A to
the n-th x is lambda.
238
00:15:13,710 --> 00:15:15,930
It's the same x.
239
00:15:15,930 --> 00:15:21,130
And every time I multiply by
A, I multiply by a lambda.
240
00:15:21,130 --> 00:15:22,920
So I get lambda n times.
241
00:15:26,220 --> 00:15:30,940
So there is the handy rule.
242
00:15:30,940 --> 00:15:33,340
And that really tells
us something about what
243
00:15:33,340 --> 00:15:35,070
eigenvalues are good for.
244
00:15:35,070 --> 00:15:39,880
Eigenvalues are good for
things that move in time.
245
00:15:39,880 --> 00:15:45,450
Differential equations, that
is really moving in time.
246
00:15:45,450 --> 00:15:51,100
n equal 1 is this first time,
or n equals 0 is the start.
247
00:15:51,100 --> 00:15:55,470
Take one step to n equal 1,
take another step to n equal 2.
248
00:15:55,470 --> 00:15:56,480
Keep going.
249
00:15:56,480 --> 00:16:01,780
Every time step brings a
multiplication by lambda.
250
00:16:01,780 --> 00:16:06,840
So that is a very useful rule.
251
00:16:06,840 --> 00:16:12,215
Another handy rule is what
about A plus the identity?
252
00:16:15,290 --> 00:16:23,870
Suppose I add the identity
matrix to my original matrix.
253
00:16:23,870 --> 00:16:25,620
What happens to the eigenvalues?
254
00:16:25,620 --> 00:16:27,290
What happens to
the eigenvectors?
255
00:16:27,290 --> 00:16:28,740
Basic question.
256
00:16:28,740 --> 00:16:33,030
Or I could multiply a constant
times the identity, 2 times
257
00:16:33,030 --> 00:16:35,820
the identity, 7
times the identity.
258
00:16:35,820 --> 00:16:39,890
And I want to know what
about its eigenvectors.
259
00:16:39,890 --> 00:16:45,260
And the answer is
same, same x's.
260
00:16:45,260 --> 00:16:47,950
Same x.
261
00:16:47,950 --> 00:16:53,010
I show that by figuring
out what I have here.
262
00:16:53,010 --> 00:16:57,630
This is Ax, which is lambda x.
263
00:16:57,630 --> 00:17:00,850
And this is c times
the identity times x.
264
00:17:00,850 --> 00:17:05,760
The identity doesn't do
anything so that's just cx.
265
00:17:05,760 --> 00:17:08,430
So what do I have now?
266
00:17:08,430 --> 00:17:16,319
I've seen that the
eigenvalue is lambda plus c.
267
00:17:16,319 --> 00:17:19,491
So there is the eigenvalues.
268
00:17:24,685 --> 00:17:30,100
I think about this as shifting
A by a multiple of the identity.
269
00:17:30,100 --> 00:17:34,830
Shifting A, adding 5
times the identity to it.
270
00:17:34,830 --> 00:17:38,780
If I add 5 times the
identity to any matrix,
271
00:17:38,780 --> 00:17:43,710
the eigenvalues of
that matrix go up by 5.
272
00:17:43,710 --> 00:17:46,800
And the eigenvectors
stay the same.
273
00:17:46,800 --> 00:17:51,650
So as long as I keep working
with that one matrix A.
274
00:17:51,650 --> 00:17:57,650
Taking powers, adding
multiples of the identity,
275
00:17:57,650 --> 00:18:02,050
later taking exponentials,
whatever I do I keep
276
00:18:02,050 --> 00:18:05,730
the same eigenvectors
and everything is easy.
277
00:18:10,910 --> 00:18:17,390
If I had two matrices, A and
B, with different eigenvectors,
278
00:18:17,390 --> 00:18:21,280
then I don't know what the
eigenvectors of A plus B
279
00:18:21,280 --> 00:18:22,330
would be.
280
00:18:22,330 --> 00:18:24,430
I don't know those.
281
00:18:24,430 --> 00:18:28,040
I can't tell the
eigenvectors of A times B
282
00:18:28,040 --> 00:18:30,610
because A has its own
little eigenvectors
283
00:18:30,610 --> 00:18:32,420
and B has its eigenvectors.
284
00:18:32,420 --> 00:18:38,310
Unless they're the same, I
can't easily combine A and B.
285
00:18:38,310 --> 00:18:47,770
But as always I'm staying with
one A and its powers and steps
286
00:18:47,770 --> 00:18:50,090
like that, no problem.
287
00:18:50,090 --> 00:18:50,720
OK.
288
00:18:50,720 --> 00:18:54,240
I'll stop there for a
first look at eigenvalues
289
00:18:54,240 --> 00:18:55,990
and eigenvectors.