Eigenvalues and Eigenvectors

MIT 18.06SC Linear Algebra, Fall 2011
Instructor: Nikola Kamburov

A teaching assistant works through a problem on eigenvalues and eigenvectors.

License: Creative Commons BY-NC-SA

[bridge]

Caution: Do not use elimination by subtracting 3rd row from 2nd row and converting the matrix in upper triangular. I thought it would be easy to read eigenvalues from the diagonal entries directly. But, this was plain stupid as that would alter the whole matrix itself and the associated eigenvalues too.

[AnupKumar-wked]

Thanks Nikola Kamburov for this amazing tutorial.

[magicbanana]

Based on what is written down, it is clear that the eigenvectors for A, and the corresponding expressions for eigenvalues based on the eigenvalues of A will work for A^2 and A^-1 - I
How do we know that these are the only eigenvectors and eigenvalues for A^2 and A^-1 - I?

det(A^2 - lambda * I) = 0 for every eigenvalue of A^2.
In the matrix (A^2 - lambda*I), lamda terms are only on the diagonal, so the determinant will be an n degree polynomial. n degree polynomial will have n zeros, so that shows that if we found n eigenvalues of A, those will also be the n only eigenvalues of A^2.

IDK how to show that those would be the only eigenvectors

[ReddipagaManimed]

Does gauss elimination preserve eigen values?

[pratikawaik]

For inv(A) - I, the eigen value 1 is not correct. Here, Nikola shows that eigen value of above matrix would be (1/lambda) - 1. So if lambda is 1 then the eigen value would be 0 which is incorrect given the matrix is invertible.

[nestorv]

how is it that A^2 v = α v and Ax = ßx have the same eigenvectors?
A (A v) = A (ß x) = α v doesnt make sense

[ZhanyeLI]

With no offence but it is difficult for this guy to elaborate the question clearly!

[yashsrivastava]

How are they writing e.vector directly. what the direct formuala for that.

[sheepfd]

For A² how do we know that A² keep the same egeinvector as A ?

[Игорь-кпи]

Nice schoolboy! I hope he will manage with math…