The problem is as follows:
Given two matrices, A and B (see below),
a) State for any a all eigenvalues for both A and B.
b) Give for both A and B the algebraic and geometric multiplicities for all of the eigenvalues.
c) For which values of a are A and B, respectively, similar to a diagonal matrix?
d) For which values of a are A and B mutually similar?
Note: By "similar" I mean as in similarity between matrices (http://en.wikipedia.org/wiki/Similar_matrix)
a) By finding the characteristic polynomial, i.e.
Is this correct? Am I missing something about the a 'complexity'?
b) Here it's simply just finding the eigenvectors and giving their number for the geometric multiplicity forand .
c) Matrix B can only be similar to the 3x3 diagonal matrix with the entries 1, -1 and 1. Thus a in A must be -1. Right?
d) What is the best way to check for matrices being similar? I know that similar matrices can be thought of as describing the same linear map with respect to different bases. Thus I've checked for rank, determinant, trace, eigenvalues, the characteristic polynomial and given the respective values for a for each of these to equal each other for A and B. Is this the way to go?
Last time you guys were able to see through the (to me) somewhat cluttered use of integrals and differentiation (I'm still very grateful for your amazing help). This time my question is similar. I'm currently trying to understand the following derivation of Lagrange's equations from Landau and Lifshi-tz' amazing first volume on mechanics:
The function for the action
What is meant by the sentence at (1)? Which calculations aren't shown here?
How is this translated to affect the variation at (2)?
How did we get to (3)?
Edit: Having studied this further the last couple of days, I've realized what a stupid question it was. I'm gonna read up on calculus of variations and understand this fully!
Just realized I might have something. Hang on.
Okay I got it.
Rewrite the original expression as
Arithmetic series wiki: http://en.wikipedia.org/wiki/Arithmetic_progression
Double counting wiki: http://en.wikipedia.org/wiki/Double_counting_(proof_technique)
First the warm-up problem:
Now to the problem:
I need to imprint the following terms on my mind:
2) Abelian groups
5) Vector spaces
If you had to explain them concretely, what would you say?
is what I've understood. Before seeing my tries at defining the terms, could you write yours? It could happen, unknowningly, that your definitions got influenced by mine in one way or the other.
I hope you can help. Getting a grasp of this should be quite important to understand linear algebra successfully.
In a physics problem, we end up with the expression
There are still a few english mathematical terms that I'm unfamiliar with. I apologize if this question is too trivial.
1. Dissect the problem and extract the important information from each sentence (this is actually two steps). This could be done as follows (some parts are underlined to add emphasis):
"A bus travels 200 km at constant speed":
"A car traveling 10 km/h faster than the bus completes the same distance in 1 hour less than the bus."
2. Set up equations:
For the bus:
3. Solve the equations.
Seemed to work out. Thanks, Bob! I really appreciate it. I can usually solve the problems, but sometimes it's good with a little confirmation that I've done it right, or getting a hint or seeing the proper solution This really, really helps, especially because I don't have many other clever heads to discuss the problems with.
Given the functionwith
Is this correct?
You guys agree? I'm mostly in trouble with the first part. Note that the vector line is just the sign of the complex conjugate, didn't know how to make it just a line.
Sinceis known to be a solution, by deflating the polynomial we get
However, deflating the 3rd degree polynomial gives
What kind of help I'm looking for: A hint to where it goes wrong in my solution. A step by step solution that I can look at to verify that my work is correct after I've used the hint. Or any other kind of help you can give me.
What kind of level I'm at: Go ahead with the hard explanations.
I have the following problem due tomorrow. I know the steps, but doing the actual computation seems tremendously hairy to me and afterwards I can't seem to get a desired result.
We are given the complex function
My current thoughts:
The third root is obviously the complex conjugate:
Method: Factorize the entire polynomium with 1 unknown rootand isolate. However, what becomes my problem is that the a's from the complex numbers won't go away, and then I have 2 unknowns.
My try at a solution:
It was used that
Where do I go from here? Or should I have taken an entirely different road?