Is this the Correct General Expression for Matrix Exponentiation?

[lapo3399]

Hi,

I have a problem which involves determining a general expression in terms of k and n for the exponentiation of a matrix of the form:

http://img242.imageshack.us/img242/2371/formwu2.png

The general expression I determined was accurate through a separate process (which does not concern this post) was:

http://img118.imageshack.us/img118/7174/genexpdf8.png

I am supposed to prove this expression is correct symbolically (ie I cannot use any examples with inputted k values) and I believe that the following should suffice:

http://img156.imageshack.us/img156/1597/prooffg7.png

Does this seem correct? Also, is there a way that I can prove this without inputting values of n (ie entirely symbolical)?

Any help is greatly appreciated.

Matt

 

[StatusX]

Try induction.

 

[lapo3399]

Unfortunately, I am in Grade 12 Discrete and we haven't done mathematical induction yet. I understand the basics of induction, but I have no idea how it should work with matrices and two variables. If anyone could help me with this, I would be grateful.

 

[shmoe]

You would only being doing induction on the one variable "n". The set-up the the same as usual,

base case: prove X^1 has that form (done)
induction step: assuming your formula for X^n is correct, prove it is correct for X^(n+1). Just use X^(n+1)=X*(X^n) and use your formula for X^n.

 

[lapo3399]

I believe this is what I am supposed to do:
Proving the expression works for X^1, assuming that it is correct for X^n and then proving it is correct for X^(n+1):

http://img301.imageshack.us/img301/6245/proof2yo4.png

However, there seems to be a problem... the last 2 is to the exponent n+1, when it should be to the power of n, shouldn't it?

 

[shmoe]

There's some funny business going on there. To the right of your "X^(n+1)=..." it looks like you have X times your expression for X^(n+1), not X^n, since it has k^(n+1)'s all over and a 2^n in front.

Then you had some problems when multiplying by this X two lines later. This may be some kind of "off by one" transcription error, but check it over carefully.

 

[lapo3399]

I see the problem... I did it almost correctly on paper, and the only thing wrong with the equations I made was that I accidentally wrote in the n+1 (which I didnt do on paper ) In any case, the real problem was that I DID substitute n+1 in for n in the exponent of the 2 when multiplying, so that error carried down.
Thanks for all the help!
p.s. Is there a way to prove it works for all negative integers too? or does this prove that?

 

[shmoe]

This only proves it for positive n. For negative integers, just invert what you have. This only makes sense when X is invertable of course.

 

[lapo3399]

So do it with n-1 instead of n+1?

 

[shmoe]

I mean if n is positive, to find X^(-n) just take your formula for X^n and find the matrix inverse.

 

[lapo3399]

I'm sorry but I don't understand - what do I do once I find the inverse? Find if it's true for n-1? And if so, assuming what?

 

[shmoe]

Once you find the inverse you are done. Invert the formula for X^n and you get a formula for X^(-n). Since you've proven your formula for X^n was true for all positive n=1,2,3,4... this gives you a formula for all negative exponents n=-1,-2,-3... in one fell swoop.

Ok, you're not quite done. It would be nice to show your formula for X^n works when n=0 as well, but that's simple enough to do. Now you will have a formula for X^n that's valid for all integers n.

 

[lapo3399]

Easy enough... Thank you shmoe, you have been most helpful :D!

 

[shmoe]

Happy to help.

By the way, good show on putting down all your work in a legible way . Since you are relatively new, you may not be aware that you can use https://www.physicsforums.com/showthread.php?t=8997" here, as in

$$X=\left[\begin{array}{cc}k+1 & k-1\\k-1 & k+1\end{array}\right]$$

(click on the image to see what you need to type to produce it).

What you did was perfectly fine though! I only point it out as another option.

 

[lapo3399]

you may not be aware that you can use https://www.physicsforums.com/showthread.php?t=8997" here

I did, but thanks for reminding me. This is part of a report for school, so I had the equations (well, most of them) previously made in that format anyway. But thanks for letting me know