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Unformatted text preview: EECS 501 COVARIANCE MATRICES Fall 2001 DEF: A random vector is a vector of random variables ~x = [ x 1 . . . x N ] . Note: Unless otherwise stated, a random vector is a column vector. DEF: The mean vector of random vector ~x is ~ = E [ ~x ] = [ E [ x 1 ] . . . E [ x N ]] . DEF: The covariance matrix K x = x of ~x is the N N matrix whose ( i, j ) th element ( K x ) ij = x i x j = E [ x i x j ]- E [ x i ] E [ x j ]. Note: K x = E [( ~x- E [ ~x ])( ~x- E [ ~x ]) ] = E [ ~x~x ]- E [ ~x ] E [ ~x ] (outer products). Also Outer product ~x~ y = [ x i y j ] = N N matrix having rank 1. Note: Inner product ~x ~ y = x i y i =scalar=Trace of outer product. 1. K x is a symmetric matrix: ( K x ) ij = x i x j = x j x i = ( K x ) ji . 2. K x is a positive semidefinite matrix: For any vector ~a , the scalar ~a K x ~a = N i =1 N j =1 a i ( K x ) ij a j 0. 3. In particular, the diagonal elements of K x have ( K x ) ii = 2 x i 0....
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