1
00:00:00,150 --> 00:00:01,040
So welcome back.

2
00:00:01,110 --> 00:00:07,410
We have previously learned why this an eigenvalue problem, and it is not really a problem because we

3
00:00:07,410 --> 00:00:13,890
can solve the eigenvalues of this matrix and these will be lambda and they are directly related to the

4
00:00:13,890 --> 00:00:15,000
eigen frequencies.

5
00:00:15,870 --> 00:00:23,130
And so we have to solve now this equation for its eigenvalues, and then we can analyze our numerical

6
00:00:23,130 --> 00:00:30,030
solution for the item frequencies and can we can figure out if we can see the ion frequencies here in

7
00:00:30,030 --> 00:00:31,020
the solution as well.

8
00:00:32,670 --> 00:00:38,550
So what we have to do is we have to calculate the eigenvalues and we can of course, do this using an

9
00:00:38,550 --> 00:00:39,480
umpire routine.

10
00:00:40,290 --> 00:00:42,690
So first of all, we must, of course, to find the matrix.

11
00:00:43,050 --> 00:00:52,920
So A is equal to NPL three, and then we can just straightforwardly write the numbers here to column

12
00:00:52,920 --> 00:00:54,720
minus one comma zero.

13
00:00:55,350 --> 00:01:07,470
And then the next line would be minus one to minus one and then zero minus one two.

14
00:01:09,390 --> 00:01:17,500
And the number high routine is NPR dot org charts.

15
00:01:17,820 --> 00:01:24,060
I think vowels and then the Matrix A. C. This is the solution.

16
00:01:24,090 --> 00:01:27,420
These are the eigenvalues of this matrix.

17
00:01:28,530 --> 00:01:35,100
So this is pretty easy, but I think it's now a pretty good time for you to exercise a bit programming

18
00:01:35,100 --> 00:01:35,790
in Python.

19
00:01:36,300 --> 00:01:43,690
Because what we have to do or what happens in this routine is that the routine solves this.

20
00:01:44,080 --> 00:01:51,150
This equation here, it solves the equation, and zero is equal to the determinant of the matrix that

21
00:01:51,150 --> 00:01:55,350
is given by a minus lambda times the identity matrix.

22
00:01:56,460 --> 00:01:59,880
So now it's your turn.

23
00:02:00,030 --> 00:02:07,440
You continue to write in this notebook and you try to define the function that, first of all, calculates

24
00:02:07,550 --> 00:02:08,370
determinant.

25
00:02:08,699 --> 00:02:14,040
Then you define another function which solves this equation here.

26
00:02:14,340 --> 00:02:21,300
So basically, it calculates the routes the zeros after is determined here, which will be a polynomial

27
00:02:21,300 --> 00:02:21,840
function.

28
00:02:22,500 --> 00:02:30,450
And then it will give you the eigenvalues that we have found here by using this very simple routine.

29
00:02:31,590 --> 00:02:34,020
So this is, of course, not really necessary.

30
00:02:34,020 --> 00:02:36,930
And if you don't want to do it, if you don't have the time, it's no problem.

31
00:02:36,930 --> 00:02:43,560
You can just skip over it or just watch my solution in the next lecture, because in the end, we will

32
00:02:43,560 --> 00:02:44,550
always use this one.

33
00:02:44,550 --> 00:02:49,770
But I think it's a good exercise and we will learn quite a lot in practice what we have learned before.