2. Let T be a mxn matrix and let v € R". Farkas's theorem asserts that exactly one of the followingholds:(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.

Answered: Image /qna-images/answer/4ca13027-5f9e-47c3-a881-862f386d69da.jpg

Answered: 2. Let T be am xn matrix and let v ER".…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter3: Matrices

Section3.7: Applications

Problem 14EQ

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Determine all nn symmetric matrices that have 0 as their only eigenvalue.
Find the general solution of the linear system (1) when A is then x n diagonal matrix A = diag[Aj, A21... ,An]- What condition onthe eigenvalues Al, ... , An will guarantee that limt_,,. x(t) = 0 for allsolutions x(t) of (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.
4. Is There A Steady State? So far, we've seen that for a conservative state transition matrix A, we can find the eigenvector, v, corre- sponding to the eigenvalue 2 = 1. This vector is the steady state since Av = v. However, we've so far taken for granted that the state transition matrix even has the eigenvalue 1 = 1. Let's try to prove this fact. (a) Show that if 2 is an eigenvalue of a matrix A, then it is also an eigenvalue of the matrix A". Hint: The determinants of A and A' are the same. This is because the volumes which these matrices represent are the same. (b) Let a square matrix A have, for each row, entries that sum to one. Show that 1 = [1 1 an eigenvector of A. What is the corresponding eigenvalue? 1]" is ... (c) Let's put it together now. From the previous two parts, show that any conservative state transition matrix will have the eigenvalue 1 = 1. Recall that conservative state transition matrices have, for each column, entries that sum to 1.
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)
Consider B = {1+x,x + x²,1 + x²}, B' = {1+x²,1 – x²,1 – 2x}. i) Show that B and B' are bases for P2. ii) Find the coordinate matrices of p(x) = -x² + 4x + 7 relative to B and B' %3D iii) Find the transition matrix PB--B'- iv) Verify that [x(p)]g' = Pg¬g'[x(p)]g-
Let A be an n × n stochastic matrix and let e be the vector in Rn whose entries are all equal to 1. Show that e is an eigenvector of AT . Explain why a stochastic matrix must have λ = 1 as an eigenvalue.
If x'=Ax +velamba t and v is an eigenvector of A, with the eigenvalue lamba also an eigenvalue of the coefficient is the form of the particular solution. lamba t matrix, then prove that vte"
5. If xy + 2xu + 3xv + uv-| = 0, 2xy + 3yu – 2xv + 2uv + 2 = 0, ſind the Jacobian matrix [ ].
5. (i) Let A be an nx n matrix over C. Show that the eigenvectors corresponding to distinct eigenvalues are linearly independent. (ii) Show that eigenvectors of a normal matrix M corresponding to distinct eigenvalues are orthogonal.
Exercise 3.1. Find f'(x), Vf(x) and V?f(x) for the following functions of n variables. (a) f(x) = }a"Hx, where H is an n x n constant matrix (not necessarily symmetric). (b) f(x) = b"Ar – a"A™Ax, where A is an m x n constant matrix and b is a constant m-vector. (c) f(x) = }E", F:(x)², where F is a vector-valued function F : R" → R" with twice- differentiable components {F;(x)}. %3D
a) Consider a matrix A with N orthonormal eigenvectors {x'} and eigenvalues {A,}. Construct the matrix S from the eigenvectors of A as follows: s- [x' x? .. x") where each column of S is an eigenvector of A. Because S is constructed from a set of linearly independent vectors, the inverse matrix S-' exists. i) State general conditions on A for it to have a set of N orthonormal eigenvectors. ii) Show that S is unitary, i.e., that S' =S-!. i) Show that the matrix A' = S'AS is diagonal, and A', = A,6,y. iv) Now consider the action of A" on a general vector y in this N dimensional space. We wish to evaluate A"y. It is proposed that we start by expanding y as y = E 4,x'. Answer the following: 1. Explain why y can be expanded as shown. Quote relevant theorems/results. 2. Find an expression for the coefficients a, in terms of a suitable inner product. 3. Show that A"y =E, \"a,x'.

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