Qualification > Math

Matrix Powers

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nid404:
Cuz we found the general form...the same thing...but we didn't know how to prove it as such....and describe it as they ask :-\

Anyway thanks a lot for your help

astarmathsandphysics:
I will do it when I get home

eightAs:
Ok I'll put forward a general induction argument. This is going to be a long post. I have an IB HL book which lists out five steps involved in an induction proof. They are:
1. Prove the result for an initial value usually (n = 1).
2. Assume the result is true for another value n = k, k>1 stating the result.
3. Consider the case n = k + 1, writing down the goal clearly - the required form.
4. Using the assumption, now show that the result is true for n = k + 1.
5. Communicate why this proves this result using mathematical induction.
There are some comments at the end of the proof.

Our conjecture is
where
Write it as:

Step 1: when n = 1. which is clearly true since A raised to 1 is A itself.

Step 2: So lets assume for some integer j, the formula is true. Notice we're not using n = k here (as described in the five steps above) as we already have a variable k.


Step 3: Substitute n = j + 1 in the original equation.

This is what we have to prove, using the assumption in Step 2.

Step 4: The ugly calculations!

Step 5: Conclude that this matrix is same as the one that appeared at the end of step 3. So the result is true for n = j + 1 when true for n = j. Since it is true for n = 1, it is true by the principle of Mathematical Induction.

Since the project is being judged for presentation, notation and whatnot. I strongly recommend that your friend sticks to the notation I used, (as they are straight out of an IB (HL) book :P ) and lays out her proof in an organized manner. :)

nid404:
omg! Thank you so very much for taking the pains to put this thing down...
THanks A Lot :D

eightAs:

--- Quote from: nid404 on January 03, 2010, 05:37:31 am ---omg! Thank you so very much for taking the pains to put this thing down...
THanks A Lot :D

--- End quote ---
No problem. By the way, do you have the markscheme for this paper?

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