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1 2.4 A Square-Root Matrix: Suppose the matrix A is a positive definite matrix. Thus, A PDP t ( PD1/ 2 Pt )( PD1/ 2 Pt ) A1/ 2 A1/ 2 , where D1 / 2 is also a diagonal matrix with diagonal elements equal to the square root of those diagonal elements of D. Therefore, A1/ 2 PD1/ 2 P t , is called the square root of A. Properties of A1 / 2 : 1. A 1 n 2 i 1 i wi wit t 1 1 A 2 A 2 (that is, A1 / 2 is symmetric). 2. 1 3. 2 A A 1 1 2 A 4. A 1 2 PD 1 2 P t n 1 wi wit , i i 1 i . 1 5. 1 D 1 / 2 is a diagonal matrix with diagonal elements equal to where 1 A. 2 2 A A 1 2 A 1 2 A 1 2 I and A 1 2 2.5 Random Vectors and Matrices: Let X ij , i 1,, n, j 1,, p, be random variables. Let X 11 X 12 X X 22 21 X X n1 X n 2 X1p X 2 p X np A 1 2 A 1 . 2 be the random matrix. Definition: E ( X 11 ) E ( X 12 ) E( X ) E( X ) 21 22 EX E ( X n1 ) E ( X n 2 ) E( X1p ) E ( X 2 p ) E X ij n p . E ( X np ) Theorem: Aln aij , B pk bij are two matrices, then E AXB AE X B . Results: E X n p Z n p E X n p E Z n p E Amn X n1 BmnYn1 AE X n1 BE Yn1 Example 11: Consider two random variables X 1 and X 2 with probability functions 0.2, x1 1 0.5, x2 0 , f1 x1 0.6, x1 0 and f 2 x2 0.5, x2 1 0.2, x 1 1 respectively. Then, E X 1 1 0.2 0 0.6 1 0.2 0 and E X 2 0 0.5 1 0.5 0.5 . Thus, for X X 1 , X 2 , t E X 1 0 E X 0.5 E X 2