16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde Vector-Matrix notation Define the vectors X and x and the mean Elr E[]=「∫(x) x1…dxn:J(x) Correlation matrix ELXX'=Xr=M M=XX o the correlation matrix arrays all the correlations among the X, with the mean squared values on the diagonal Note that. The correlation matrix is symmetric If X; and X; are independent, the correlation is the product of the means (from the product rule for the pdf) XX=XX Covariance matrix [(r-XXX-x)]=ELXXJ-E[X]X-XELr]+X E[ XX-XX XX-XX=C C,=[(X-X)(X-X) XX-XX This is by definition the covariance between Xi and X. So the covariance matrix arrays all the covariances among the X, with the variances along the diagonal. Page 5 of 916.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Vector-Matrix Notation Define the vectors X and x and the mean E[X ] . ∞ ∞ E X dx1... dx xf ( x) [ ] = ∫ ∫ n n −∞ −∞ ⎡ ⎤ ∞ ∞ x1 dx1... dx ⎢ ⎥ M f ( x) = ∫ ∫ n n ⎢ ⎥ −∞ −∞ ⎢ ⎥ x⎣ ⎦n = X ⎡ X ⎤1 ⎢ ⎥ = M ⎢ ⎥ ⎢X ⎥ ⎣ n ⎦ Correlation Matrix T E ⎡XX T ⎤ ⎦ ⎣ = XX = M M ij = X Xi j So the correlation matrix arrays all the correlations among the Xi with the mean squared values on the diagonal. Note that: • The correlation matrix is symmetric. • If Xi and Xj are independent, the correlation is the product of the means (from the product rule for the pdf). XiX j = X Xi j Covariance Matrix T T T T T E ⎡( X − X )( X − X ) ⎤ = E ⎡XX ⎤ − E X ] X − XE ⎡X ⎤ + XX ⎣ ⎦ ⎣ ⎦ [ ⎣ ⎦ T T = E ⎡XX T ⎤ ⎦ − XX − XX T + XX ⎣ T = XX T − XX ≡ C T Cij = ⎡( X − X )( X − X ) ⎤ ⎣ ⎦ij = X Xi j − X Xi j This is by definition the covariance between Xi and Xj. So the covariance matrix arrays all the covariances among the Xi with the variances along the diagonal. Page 5 of 9