Use the matrices P and D to construct a spectral decomposition of A = PDP - 1 9 0 O A= 0 : P = [ U , U2 Us] = NIm - Im NIM NIM NIM - IM - Im NIM NIM D=03 O 0 O A spectral decomposition of A is A = (Du, u, + (Du,u) + (Du,uj, where u, u! = . uzu) =, and ugu] =] (Simplify your answers.)-12/02 3. [5 points] Suppose that the number of customers arriving at the Woori Bank follows a Poisson process with a rate of Aw = 0.5 per minute, the number of customers arriving at the Hana Bank follows a Poisson process with a rate of An = 0.2 per minute, and the two Poisson processes are independent. Let( We )be the waiting time to have the kth customer at the Woori Bank and /, be the waiting time to have the kth customer at the Hana Bank. Find the probability distribution of the random variable Wa H 2 and explain why.Eonsider the birth-and-dealh process with the following mean rates. The birth rates are in: 2, 11: 3, In: 2,. 1g: 1, and}. =forn: 3.11142de rates are [11:3, [11:4, |.J.a= 1, and p." =2forn :- 4. a] Construct the rate diagram for this birthanddeath process. B] 1 Construct the rate diagram for this birthand death process. cl] Solve these equations to nd the stead ystate probability distribution PD, P1 ..... d] Use the general formulas for the birthanddeath process to calculate PU, P1 ..... Also calculate L , Lq , W, and No. Implement Jacobi method using any programming language Implement Jacobi method using any programming language for using the following initial values 12x, + 3x,- 5x, = 1 x + 5x2 + 3x, = 28 3x + 7x + 13x3 = 76 X = 0 XProblem 4.4 of Sparke & Gallagher: Show that the Sun exerts a greater gravitational force on the Moon than the Earth does. Calculate the Jacobi (or Hill, or Roche!) radius for the Earth-Sun system, and show that the semimajor axis of the Moon's orbit is smaller than the Jacobi radius this is why the Moon remains bound to the Earth