Solving Linear Dynamical Systems with Eigenvectors and Eigenvalues

[himurakenshin]

How would i solve such a question?
in a system V sub(k+1) = A V sub(k)
approximate Vsub(k).

there is an equation V sub(k)=b1 (lambda)^k X1
but I don't know what b is

 

[matt grime]

what is A, what is v_r, what is lambda, and what is b1, not to mention X1

 

[himurakenshin]

sorry :) I believe in this equation, lamba is the eigen value, X_1 is the eigen vector and I don't know what b1 is.
The question is I have been given the linear dynamical system V_k+1 = AV_k and they have given values for A and V_0 and I have to approximate V_k

 

[matt grime]

So A is a matrix, nxn is it? Every nxn matrix satsfies its own characteristic polynomial.

You really can't just write out your homework problems and expect us to know what the notation means.

 

[himurakenshin]

sorry about that. I just don't understand how to solve such a problem.

 

[matt grime]

so this relates to an entirely separate post? i also don't know what you mean by approximate in this context.

 

[himurakenshin]

sorry my last post was wrong I edited it. OK I will restate the question:
there is a linearly dynamical system V_k+1 = AV_k
I have the values of A and V_0
The have asked me to approximate the value of V_k.

That's the question. I just don't understand how to do it

 

[matt grime]

This would depend on the situation. If A has a complete set of eigenvectors, that is theyform a basis, then we can do it in terms of eigenvectors.

the r'th power of an nxn matrix can be written in terms of the first n powers, but really we can only suggest things since we have no idea what you're doing. Like I say, what does approximate even mean? In what norm? Is A even an nxn matrix? Why not an infinite dimensional dynamical system?

 

[himurakenshin]

they say to approximate using (i believe) this equation

V_k = b_1 * lamba^k * X
where lambda is the eigen value of A and X the eigen vector of A (I am presuming)

 

[matt grime]

so what, b_1 is a real number, another matrix (I'm assuming we're talking nxn matrices for A, but if you wouldn't mind saying so...) well, if v_k is indeed that, then it is a multiple of an eigenvector if i remember correctly, and Aw=tw for any eigenvector w with eigen value t so A^k(w)=t^k(w), right?

 

[himurakenshin]

um... b1 is a real number but I don't know how to get it. Do you know of anywhere on the net where I can read up on these stuff ? I can't seem to find anything in my textbook

 

[matt grime]

well, maybe you ought to find out? please don't take that the wrong way, but you've asked for help on a question, and you don't know the meaning of the symbols involved, and haven't even confirmed that we're talking about applying an nxn matrix to a finite dimensional vector space or anything. I think the problem is that you don't know the basics, not that you can't do the question. look at the definitions of allthe things in your notes, or whatever book you're reading again, and again, and again until you know them by heart, then try the question again. perhaps reread examples like it in the notes/book. then when you know what b1 is perhaps you'll be able to do the question.

you're best bet for getting an answer here is to put the question Wrod for word in a post and explain all the meanings of all the symbols and all the assumptions (liek A is a linear map from a finite dimensional vector space to itslef or whatever it is, cos yo've still not said).

 

[himurakenshin]

I pretty much did do a word to word post of the question and yes i forgot to mention that A is an nxn matrix (its a 2x2)

 

[matt grime]

word for word AND explain what all the undefined symbols are...

is A an actual matrix with explit entries? what are its entries? is v an actual honest to god vector with two real entries like 3 and 7? I mean if A is diag{2,3} and v_0 is e_1+e_2 and if X=e_1 then I can't remotely approximate the forward orbit of V using X, can I?

 

[himurakenshin]

Ok could you please explain what a "linear dynamical" system is and this one of the questions,
A= 3 -2 V_0 = 3
2 -2 -1

approximate V_k

 

[matt grime]

you should be aware that what you write in the white box doesn't appear in the forum like that, there's a thread in physics explaining latex.


so you didn't post the question verbatim then with afull explanation? sorry, it's late and i don't like being jerked around.

A's char poly is (t-3)(t+2) + 4 = t^2-t-2, i think which has roots 1 and -2, right? So write V as ru+sw where r and s are real numbers and u and w are the eigenvectors, obviously only one of those is important in the long term behaviour since Aw=w, and Au=-2u, say.