Homework 4, due February 10, 2010

[goodwine]

Reading: all of chapter 6 in the course text.

Exercises: from THIS HANDOUT (not the purchased book), problems 5, 6, 7, 10 and 11.

For problem 10, the inverse of T is equal to the transpose of T only if you normalize the eigenvectors to have unit length.

[goodwine]

On question 5.4/in general after computing (A-lambda I) are you allowed to perform row operations before squaring/cubing the matrix to get the eigenvectors? Thanks.

That's a good question and I don't know the answer.

On the one hand, I don't think that row operations and matrix multiplication are operations that can have their order changed, i.e., the operations don't commute.

On the other hand, I am inclined to think that since the purpose is only to find the null space, I'm inclined to think yes.

At this point my advice would be to say not to risk it, but I'd be interested in the answer if you find out.

[elegault]

For problem 11, the question asks for us to compute the matrix exponential for A1, A2, and A9. Then i think it says to verify the solutions for A3 and A9. Is that supposed to be A2 not A3? or should we compute A3 as well? Thank you!

[goodwine]

For problem 11, the question asks for us to compute the matrix exponential for A1, A2, and A9. Then i think it says to verify the solutions for A3 and A9. Is that supposed to be A2 not A3? or should we compute A3 as well? Thank you!

You can use A2 for the last problem.

[tmo3290]

For problem 11, since we never did Exercise 2, should we still do the part where it asks us to verify the solutions?
And if so, did you mean A2 and A9, rather than A3 and A9, since those are the ones the first part of the problem asks us to do?

[tmo3290]

sorry, I should refresh the page more often.

[goodwine]

For problem 11, since we never did Exercise 2, should we still do the part where it asks us to verify the solutions?
And if so, did you mean A2 and A9, rather than A3 and A9, since those are the ones the first part of the problem asks us to do?

I didn't do that on purpose, but yes, verify the solution. That should be easy for A2 since you did that matrix on homework 2.

[goodwine]

For the first problem in the fourth homework set, the last two matrices are almost identical. I got the exact same generalized eigenvectors, and that will always make me nervous on assignments or tests. Obviously there will be a difference that shows up in the equation for the general solution but is this all you were trying to show by switching around a few matrix elements?

The 1s that are on the off diagonal significantly alter the solution. It is possible, however, to have the exact same generalized eigenvectors. When they get multiplied through the powers of (A - lambda I) there should be some differences. In particular, how many t's are in the answer may be different.

[AL089]

Can you use generalized eigenvectors in the T matrix when trying to diagonalize A?

[goodwine]

Can you use generalized eigenvectors in the T matrix when trying to diagonalize A?

If you do, it's close to, but not exactly, diagonal. Having said that, I don't remember assigning a problem that would require you to do that.

[ctalley1]

Professor, could you please post the solutions to HW4?

Thanks,
Chuck

[goodwine]

Professor, could you please post the solutions to HW4

Done.