6.4 Suppose the vector z = (x, y), where x (p×1) and y (q ×1) are jointly distributed with mean vectors μx and μy and with covariance matrix

Consider projecting x on M= sp{1, y}, say, x = b + By.

(a) Show the orthogonality conditions can be written as E(x − b − By) = 0, E[(x − b − By)y ] = 0, leading to the solutions

(b) Prove the mean square error matrix is

(c) How can these results be used to justify the claim that, in the absence of normality, Property P6.1 yields the best linear estimate of the state xt given the data Yt, namely, xtt , and its corresponding MSE, namely, Pt t ?

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cov(2) =