The quiz in the video ‘Eigenvalues and eigenvectors’, asks us to find the eigenvalues and eigenvectors of the matrix [[9,4][4,3]].
I understand that the eigenvalues are 11 and 1 after solving the quadratic equation.
However, when I try to find the eigenvectors using the method describe in the video:
For λ=11
9x + 4y = 11x => 4y = 2x => 2y = x
4x + 3y = 11y => 4x = 8y => x = 2y
So I cannot find a unique point.
For λ=1
9x + 4y = x => 4y = -8x => y = -2x
4x + 3y = y => 4x = -2y => 2x = -y
So again I cannot find a unique point.
Also, the solution in the video says that the eigenvectors are (2,1) and (-1,2). When I input the matrix to an online calculator (for example here: Eigenvalues and Eigenvectors), then I get eigenvectors (2,1) and (-0.5,1).
Well, notice that the vectors (-1, 2) and (-0.5, 1) point in the same direction, right? One is just twice as long as the other. The eigenvectors just express a direction. I have not taken M4ML, so I don’t know what they say in the course materials but one standard approach is to normalize the eigenvectors to have length 1 (2-norm of 1). It might be worth listening again to what the lectures say with what I just said in mind. What (if anything) do they say about the lengths of eigenvectors?
Thank you, I understand this know. To confirm, when I go to find the eigenvectors for eigen value λ=11 I get these two equations:
x = 2y
x = 2y
I don’t need to find a unique point from these two equations, I just need to find one vector that satisfies them which will give me the direction of the eigenvector. Is this correct?
Well, as I mentioned in my earlier reply, I have not taken M4ML so I don’t know what they say in the course materials about how they are handling the length of eigenvectors. But I was a math major back in the day and what you say is correct from a mathematical point of view. Some math libraries will automatically normalize the eigenvectors to have length 1, but that is not technically necessary.
The following equations:
2y=x
x=2y
have more than one solution.
One solution is: x=1 & y=0.5
Another solution is: x=2 & y=1
Another solution is: x=4&y=2
To find these you just need to substitute one equation into the other, then you have 2y=2y. Then you specify whatever value you want for y, and then you find the corresponding value for x.
The point here is that you don’t need to find a unique point from the two equations. You just need to find one solution, for example x=1 & y=0.5. Then this vector (1,0.5) gives you the direction which is what you want. (It has the same direction as the vector (2,1) for example)
You can’t get a unique solution for that system of two equations, because they’re exactly the same equation. So you have one equation and two unknowns.