Let X Є pxn denote a zero mean observation matrix matrix of obser- vations, P be a pxp orthogonal matrix, and Y = PTX. (a) Show that if C is the covariance matrix of X then PTCP is the co- variance matrix of Y = PT CP. (b) It can be verified that tr (FG) = tr (GF) for any two n x n matrices F and G. Use this fact to show that the total variance of the data in Y is equal to the total variance of the data in X.

The covariance-variance matrix of a set of observations, xx, is a symmetric matrix. The Covariance matrix of functions of these observations should also be symmetric as the two covariance matrices are related by the equation Eyy= JEXXJT. Using laws on matrix transposition, prove that, indeed, Eyy is symmetric by showing that the product JEXXJT is symmetric, i.e., the original product and its transpose are equal. You need to list the transposition law concerned, or a property of a matrix, at every step of the proof.

1. Generate random 100 MVN data, p=20. The covariance matrix should be positive semi- definite symmetric matrix. (1) Show the generated data matrix. (2) Plot the scatter plot to show the correlation among variables.

Define Covariance matrix.

Suppose you have obtained from SPSS the correlation matrix in Appendix 1. According to the information in this matrix, which two variables exhibit the strongest linear relationship?

Which of the following correspond to the adjacency matrix?

Estimate OLS estimators and Their variance Via Matrix notation

Consider the following two data vectors: A = [3,7,11]B=[3,2,1] Calculate the covariance matrix of these two vectors.

Find the adjacency matrix

Given data points (2,-1), (2, 1), (3,2), (0,-1), and (3,-1), compute the sample covariance matrix. Record the entries of covariance matrix row by row as decimal numbers, separated by commas.