Could you please help with question 261 and 262? Thanks

5.3 Difference Equations So what can we do with spectral decompositions that we could not do with the SVD? We have already seen examples, like the Stein equation, which can be more efficiently solved via spectral decompositions. However the classical examples are infinite sets of equations where spectral decompositions (for now at least) are the only way. Let u[n] E C for n = 0, 1, 2, ..., be a sequence of unknown column vectors that satisfy the constraints un + 1] = Au[n] + f[n], (5.3) where A E C VAN and f E C and are both known quantities. The question is to find all sequences un] that satisfy the above constraints. Exercise 258 Write the above set of equations in the form Fx = b. Note that there are an infinite number of unknowns and equations. So, even though the constraints are linear equations it is not easy to develop a procedure like Gaussian elimination to find the solutions. Fortunately it turns out that a spectral decompo- sition of A is sufficient. The idea is to first figure out the nullspace of the associated matrix. Consider the so-called homogenous equations un[n + 1] = Aun[n], n 20. It is clear that the only solutions are of the form un [n] = A" un [0]. From this we can guess that a solution of the equations is Up(n + 1] = EAn- kf[k], K = 0 assuming up[0] = 0. Exercise 259 Verify that up does indeed satisfy the difference equation 5.3. Therefore the general solution is u[n] = Anu[0] + An-if [0] + An-2f[1] + . . . + Aof[n - 1]. Exercise 260 Verify this.This formula is a bit cumbersome to use. A simplification is available via the Jordan decomposition A = VJV-1. Exercise 261 Show that A" = VJ"V-1. Remember that J is block diagonal with each diagonal block of the form Al + Zip. Therefore we only need to figure out a formula for (XI + Zp)". (Why?) Exercise 262 Prove the binomial theorem ( at b ) " = I (2 ) akon - k for a, be C. Exercise 263 Show that if AB = BA then (A + B) " - > (2AB-K. Exercise 264 Show that (AI + Zp)" is an upper triangular matrix with n! An -k (n - k)!k! as the entry in the k-th super-diagonal. So 1" is the entry on the main diagonal, for example. Exercise 265 Using the Jordan decomposition develop a simple formula for V-lu[n], the solution of the difference equation in terms of V-If