2. Let T be am xn matrix and let v ER". Farkas's theorem asserts that exactly one of the following holds: (i) there exists x Rm such that x ≥ 0 and XT = v, € (ii) there exists y ER" such that yv' < 0 and Ty' ≥ 0. Use this to prove that a finite stochastic matrix has a non-negative non-zero left eigenvector corre- sponding to the eigenvalue 1.

Consider a birth and death process with states {0,1}. Let the birth and death rates be Ao = a > 0, A₁ = 0, μo = 0, and µ₁ = b > 0. = ko = A1 (a) Show that the infinitesimal generator matrix A = [ a a satisfies the property A² = - 2 -(a + b)A. b Thus, for n = 1, 2, 3,..., A = (a - b)-1A holds. n (b) Using the result from part (a) to compute the matrix exponential et in closed form and thus find the transition probability matrix P(t) = e¹A, t≥ 0. (c) Compute lim →∞ P(t) and use it to find the stationary distribution.

1-α B Show that every 2 x 2 stochastic matrix has at least one steady-state vector. Any such matrix can be written in the form P = α 1-B where x and ẞ are constants between 0 and 1. (There are two linearly independent steady-state vectors if a = ß= 0. Otherwise, there is only one.) If P is a stochastic matrix, then a steady-state vector, or equilibrium vector, for P is a probability vector q such that Pq=q. If a =B=0, then P= 10 01 Notice that Px = x for any vector x in R². Determine two linearly independent steady-state vectors in this case. 0 (Use a comma to separate answers as needed.)

4. Let A be an n x n matrix such that A31 = 0n. (a) Show that In - A is invertible. and a (b) Let B be an invertible n × n matrix. Is B – A invertible? Provide a proof if yes, counterexample otherwise.

Let Y Z3+ e denote the standard form of the multivariate regression model where Y is a matrix of n observations on m dependent variables, Z is the nx (r + 1) design matrix, 3 is the matrix of regression coefficients and denotes the error matrix. Let Σ denote the m x m covariance matrix of any row of € and assume the rows are independent. The least square estimate for 3 is 3 (Z'Z)-¹Z'Y. The projection matrix is H = Z(Z'Z)-¹Z'. Let the ith column of 3 be denoted by 3, and the i, jth element of Σ by oij. Define the sample covariance matrix of the residuals by Prove Σ = 1 n-p-1 (c) E(ê) = 0 (d) cov(ei, k) = Oik (I-H)

2.16. Consider an arbitrary n × p matrix A. Then A'A is a symmetric p x p matrix. Show that A'A is necessarily nonnegative definite. Hint: Set y %3D Аx so that y'y — х'А'Ах.