For a Markov matrix, the sum of the components of x equals the sum of the components of Ax. If Ax = AX with,\ cf= 1, prove that the components of this non-steady eigenvector x add to zero.

Suppose you are told that the maximum eigenvalue of matrix A is dmax=3. Write a java program to compute each node's eigenvector centrality based on the equation c,(v;) =E; AjiC, (v;) Adjacency matrix ( for A) Adjacency Matrix 3 4 1 1 1 1 1 2 3 1 1 1 4 1 1 1 6 1 1 7 1 1 1. 1. 1. 00

Prove that the sum of two negative definite matrices A and B is negative definite (A+B) meaning that for matrix (A+B) all eigen values will be negative or have negative real parts

A 2x2 matrix has trace -13. One of its eigenvalues is -9. What is the other eigenvalue?

(6) Let Q1 be an m x n matrix with orthonormal columns and Q2 be an n x p matrix with orthonormal columns. Show that Q1Q2 has orthonormal columns.

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.