3 Problem 3 Suppose there are three birds flying in the sky and assume the birds' speeds are v, (k), v2(k) and vs(k) at moment k, respectively. Each bird will adjust its speed based on the average of the other two birds' speeds, i.e. at moment & + 1, the birds' speeds are vi(k + 1) =2 12 (k) + us(k) U2(k + 1) - Pi(k) + us(k) and vs(k + 1) _ "i(k) + va(k) 2 2 2 respectively. Assume the initial speeds are v1 (0) = 8m/s, v2(0) = 9 m/s and v3(0) = 13 m/s. 1. What are the speeds of the birds at moment k = 1? 2. Assume we lump the speeds into a vector as r(k) = [u (k), v2(k), us(k)]" e R3, then the relation between r(k + 1) and r(k) can be described as r(k + 1) = Ac(k) with A e R3x3. Find the A matrix. 3. Find a canonical diagonal matrix similar to A, and give the orthogonal transformation matrix. 4. Find the speeds of the birds when k -+ co, i.e. lim r(k). k-+0o