Thread: characterristic equation

there're some problems about sequence of number that are solve by using the characterristic equation, but i don't full understand the principle. for e.g : It is said that:
. a sequence of number X(n+2)=C1X(n+1)+C2X(n)
given r，s then X(n+2)-rX(n+1)=s[X(n+1)-rXn]
so X(n+2)=(s+r)X(n+1)-srXn

C1=s+r
C2=-sr
then eliminate s , then we got the so-called characterristic equation will be : r*r-C1*r-C2=0

I just don't understand that how can the 'S' can be eliminated, since X(n+2), Xn and X(n+1) can' t be combined?
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C₁ = s + r
C₂ = −sr

Multiplying the first equaiton by r gives

C₁r = sr + r²

Now add this to the second equation. That’s how you eliminate s.

No , i mean where are the Xn, X(n+1)...?

I don't understand your argument, but I know what you're trying to solve. So suppose we have a recurrence relation:

x_n+1 = c₁x_n+c₂x_n-1

We can describe this by a matrix equation:



Then the characteristic polynomial of the 2x2 matrix is t²-c₁t-c₂, and this helps us express x_n as a sum of exponentials. Namely, let r_i, i = 1,2, be the roots of the characteristic polynomial. Then it's not too hard to see that, if the roots are distinct, we have that:



are the eigenvectors of the matrix, each with eigenvalue r_i. (If the roots are not distinct, then there's a unique eigenvector, which looks the same as above, but this implies the matrix is not diagonalizable, which makes things a little murkier, so let's ignore this case.) So express:



And this gives you:



In particular, you have x_n = a₁r₁ⁿ + a₂r₂ⁿ.