Bill Goodwine's Courses

Reading: Chapter 6 through 6.4

Exercises: 6.1 (A_1 only), 6.2 (A_7 only), 6.4 (A_1 and A_3 only) and 6.5 (A_5 and A_6 only).

Do this instead:
Exercises: 6.1 (A_1 only), 6.4 (A_1 and A_3 only) and 6.5 (A_7 and A_8 only).

Professor,

I was working through matrix A_7 on 6.2 and kept getting eigenvalues of 1,2,8, and 8, but according to the problem description the matrices should either abide by Theorum 6.1 or 6.2. After confirming the eigenvalues with MATLAB, it seems the matrix doesn't follow either because of the repeated roots and it is not symmetric. Is it possible to use the repeated roots method to find an additional linearly independent eigenvector or is it a typo or mistake on my part? Thanks.

mwilcox3 wrote:Professor,

I was working through matrix A_7 on 6.2 and kept getting eigenvalues of 1,2,8, and 8, but according to the problem description the matrices should either abide by Theorum 6.1 or 6.2. After confirming the eigenvalues with MATLAB, it seems the matrix doesn't follow either because of the repeated roots and it is not symmetric. Is it possible to use the repeated roots method to find an additional linearly independent eigenvector or is it a typo or mistake on my part? Thanks.

Both theorems allow you to know beforehand if you will be able to find a full set of linearly independent eigenvectors, but even if a matrix satisfies neither, it may still have a full set. Mathematically, they provide sufficient, but not necessary, conditions for a full set of linearly independent eigenvectors.

mwilcox3 wrote:Professor,

I was working through matrix A_7 on 6.2 and kept getting eigenvalues of 1,2,8, and 8, but according to the problem description the matrices should either abide by Theorum 6.1 or 6.2. After confirming the eigenvalues with MATLAB, it seems the matrix doesn't follow either because of the repeated roots and it is not symmetric. Is it possible to use the repeated roots method to find an additional linearly independent eigenvector or is it a typo or mistake on my part? Thanks.

Ok, I had a chance to fully work it out and you were doing the correct computations. See the email I just sent to the whole class. My answer above is still correct, but doesn't apply to this problem.

Depending on how we reduce our matrices, is it possible to get multiple different eigenvectors for a given complex eigenvalue?

sprender wrote:Depending on how we reduce our matrices, is it possible to get multiple different eigenvectors for a given complex eigenvalue?

Without repeated eigenvalues, the only possible difference between eigenvectors is that they may differ by a scalar multiple. However, for complex ones it can sometimes be VERY hard to tell they differ by only a multiple because that multiple can be a complex number. Consider \xi = a + i b where a and b are the real and imaginary components of the eigenvector and multiplying it by something like 4+7i. It would be hard to tell the one that was multiplied was related to the starting one.