Hi all, I can’t find the eigenvectors in question 7. I’m stuck after getting the eigenvalues of 2 and 1. I’m stuck when I try to solve the eigenvector for eigenvalue of 2,

For Eigenvalue = 2,

2 0 0 x

1 2 1 y = 2 . x y z

-1 0 1 z

I get the following equations from the above matrix using Luis’ method from the 2x2 matrix example in Week 4 lectures notes (slide 324),

x = x

x = -z

-z = x

How do I derive the eigenvector for eigenvalue of 2 in this case?

Hi @Roger_S!

Sorry for the late response!

You are almost there. Your calculations are correct. You have found that the eigenectors (x,y,z) related to \lambda_2 = 2 must satisfy x = -z (or -x = z, it is the same).

Therefore, the eigenvectors are of the form (-z,y,z). Therefore, you can write (-z,y,z) = (-z,0,z) + (0,y,0) = z(-1,0,1) + y(0,1,0). so any eigenvector can be expressed as the sum of the vectors (-1,0,1) and (0,1,0). So you can pick both vectors as eigenvectors.

If you have any further help, let me know.

Thanks,

Lucas

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I’m struggling with this question. I can’t even find the eigenvalues for the matrix. This is a 3x3 matrix, but I believe all the lectures covered 2x2 matrices. Why are we quizzed on material or techniques that were not covered? Or maybe I’m misunderstanding how to solve this?

For 2x2s, we use the characteristic polynomial formula to find the roots and get the eigenvalues. But for 3x3, are we required to solve a cubic polynomial of λ^3 + 5λ^2 - 8λ +4 to get the roots? I don’t know how to do that by hand, and neither does ChatGPT, lol

This is the only answer I have found that explains the correct choice in the quiz for this question. If it helps someone for a scenario where the λ is repeated when trying to find eigenvectors for the repeated λ, you should watch this YouTube video:

The problem, once again, with the Coursera quiz question is that the technique shown in the above YT video also does not prove helpful because of the nature of the original matrix in question, so the answer provide by @lucas.coutinho is the only one that makse sense. Thank you so much to Lucas.

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Hi @Jim_Beno!

First, thanks for your feedback. The result we present for 2x2 is the general result: it can be done with any n \times n matrix, but you do have a point that we are quizzing questions we do not explicitly cover in lectures! I will adjust it.

You are right in your calculations. When computing the characteristic polynomial, you find that the polynomial is (1-x)(2-x)(2-x) (only the principal diagonal will have non zero values!). So the roots (1-x)(2-x)(2-x) = 0 are just 1 and 2 (twice).

And yes, usually it is quite hard to find roots of polynomials! I would recommend using a mathematical tool, such as WolframAlpha to get these kind of computations. ChatGPT isn’t a math engine, but a NLP one, so you must not rely into its answers.

Here is WolframAlpha’s solution.

I hope that helps, and again, thanks for your feedback!

Thanks,

Lucas