Determine the eigenvalues

2. (15 points) Let A = 2 _2] S (1) (5 points) Determine the eigenvalues of A (2) (5 points) Determine the eigenvectors of A (3) (5 points) Write the spectral decomposition of AWhich of the following is an advantage of multivariate correlational research over bivariate correlational research? O Bivariate correlational studies help address internal validity, whereas multivariate correlational studies do not. Multivariate correlational studies establish covariance, whereas bivariate correlational studies do not. O Some multivariate correlational studies help to address temporal precedence, whereas bivariate correlational studies do not. Multivariate correlational studies help to address external validity, whereas bivariate correlational studies do not.2. A Markov chain with state space {1, 2, 3} has transition probability matrix 00 0.3 0.1 a: 0.3 0.3 0.4 0.4 0.1 0.5 (a) Is this Markov chain irreducible? Is the Markov chain recurrent or transient? Explain your answers. (b) What is the period of state 1? Hence deduce the period of the remaining states. Does this Markov chain have a limiting distribution? (c) Consider a general three-state Markov chain with transition matrix 3011 3012 1013 P = P21 P22 P23 1031 P32 P33 Give an example of a specic set of probabilities jag-'3; for which the Markov chain is not irreducible (there is no single right answer to this1 of course l]. 4. Consider a discrete-time Markov chain with the following probability transition matrix 0 0 0 0 7 1-3-y P = 1-I-VVO T 0 0 1 0 Is it possible to choose values for a and y so that the Markov chain has the following properties? In each case, state the values of a and y, or give a brief reason why it is not possible. (a) The Markov chain has period 2. [2) (b) The Markov chain is reducible. (c) The Markov chain has at least one transient state. UNN (d) The Markov chain has invariant distribution (1/4, 1/4, 1/4, 1/4)