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9.1. Let x and y be p × 1 and q × 1 random vectors, respectively, and let Σ = Cov ( x ) = (² Exx Exy Σyx Σyy as in Secti
Posted: Wed May 04, 2022 12:03 pm
by answerhappygod
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9.1. Let x and y be p × 1 and q × 1 random vectors, respectively, and let Σ = Cov ( x ) = (² Exx Exy Σyx Σyy as in Section 9.1.2. Consider the problem of maximizing a¹ Exyb subject to a¹ Exxa = b² Σxxb = 1. (a) Using the Lagrange multipliers ◊ and µ for the two constraints, show that 0 = µ and that is a solution of the equation -0xx Exy det = 0. (9.83) (b) Show that for any nonsingular square matrices C and D, det (-DCXXXC™_CXDD T = 0, DEyx CT and therefore the solutions of (9.83) are invariant with respect to the transformation x = Cx, ỹ = Dy. Exercises 235 (c) Show that if is a solution of (9.83), then 0² is an eigenvalue of Σχ. Σ.. Στ. Σyx and is also an eigenvalue of Στ. Σ.Σ.Σ. xy yy yy yx