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Ad123q

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Was wondering if anyone could help me with this topic, which relates to the solution of simultaneous linear difference equations and diagonizable matrices.

Sorry for the lack of latex, but as such I will denote an exponent by "^" and a subscript by "_".

Say we have a system of simultaneous linear difference equations

x_n+1 = ax_n + by_n

y_n+1 = cx_n + dy_n

where a, b, c, d are numbers, and "x_n+1" etc is "x subscript n+1".

Denote v_n by the column vector (x_n , y_n) and let the coefficients a, b, c, d form a matrix

A= a b

c d

Then we can write v_n+1 = Av_n.

The solution of the system is then meant to be, in general, v_n = A^nv_0,

where A^n is matrix A raised to power n, calculated by considering that P^-1AP = D (diagonal) and so A^n = PD^nP^-1,

where P^-1 is the inverse of the matrix P, viz the matrix formed by eigenvectors.

Thanks very much for any help in advance.