Transcription of Properties of the Covariance Matrix
1 Properties of the Covariance MatrixThe Covariance Matrix of a random vectorX Rnwith mean vectormxisdefined via:Cx=E[(X m)(X m)T].The (i, j)thelement of this Covariance matrixCxis given byCij=E[(Xi mi)(Xj mj)] = diagonal entries of this Covariance matrixCxare the variances of the com-ponents of the random vectorX, ,Cii=E[(Xi mi)2] = the diagonal entries are all positive the trace of this Covariance Matrix ispositive, ,Trace(Cx) =n i=1 Cii> Covariance matrixCxis symmetric, ,Cx=CTxbecause :Cij= ij= ji= Covariance matrixCxis positive semidefinite, , fora Rn:E{[(X m)Ta]2}=E{[(X m)Ta]T[(X m)Ta]} 0E[aT(X m)(X m)Ta] 0,a RnaTCxa 0,a the Covariance matrixCxis symmetric, , self-adjoint with the usualinner product its eigenvalues are all real and positive and the eigenvectors thatbelong to distinct eigenvalues are orthogonal, ,Cx=V VT=n i=1 i~vi~ a consequence, the determinant of the Covariance Matrix is positive, ,Det(CX) =n i=1 i eigenvectors of the Covariance Matrix transform the random vector intostatistically uncorrelated random variables, , into a random vector with adiagonal Covariance Matrix .
2 The Rayleigh coefficient of the Covariance matrixis bounded above and below by the maximum and minimum eigenvalue : min aTCxaaTa,a R
