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## Forming the co-variance matrix

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**Forming the co-variance matrix**of the data Multiplying both sides times fik, summing over all k and using the orthogonality condition: Canonical form of eigenvalue problem eigenvectors eigenvalues**I is the unit matrix and are the EOFs**Eigenvalue problemcorresponding to a linear system:**Matrix = [6637,18]**rows > columns**Matrix ul = [6637,18]**>> uc=cov(ul); >> u1=ul(:,1); >> sum((u1-mean(u1)).^2)/(length(u1)-1) ans = 9.6143 >> u2=ul(:,2); >> sum((u1-mean(u1)).*(u2-mean(u2)))/(length(u1)-1) ans = 10.1154**Covariance Matrix**Maximum covariance at surface**>> uc=cov(ul);**>> [v,d]=eig(uc); eigenvalues (or lambda) >> lambda=diag(d)/sum(diag(d));**>> uc=cov(ul);**>> [v,d]=eig(uc);**>> uc=cov(ul);**>> [v,d]=eig(uc); >> v=fliplr(v);**Mode 2**13.2% Mode 1 85.3%**>> ts=ul*v;**ts=[6637,18] Mode 1 85.3% Mode 2 13.2% Mode 2 13.2% Mode 1 85.3%**>>for k=1:nz**vt(k,:,:)=ts(:,k)*v(:,k)'; end vt=[18, 6637,18] mode # evolution in time time series # >> v1=squeeze(vt(1,:,:))’; >> v2=squeeze(vt(2,:,:))’; Depth (m)**Complex Empirical Orthogonal Functions – James River Data**u v Linear combination of spatial predictors or modes that are normal or orthogonal to each other**Rotated 49 degrees**Streamwise Cross-stream**Mode 1**96.5% Mode 1**Mode 2**2.5% Mode 2**Streamwise**Mode 1 96.5% cross-stream**Mode 1**96.5% Principal-axis Mode scaling cross-axis**Mode 2**2.5% Streamwise Mode scaling Cross-stream**Mode 2**2.5% Streamwise Mode scaling cross-stream**Mode 1**75% m/s m/s m/s**Mode 2**22% m/s m/s m/s**Phase of EOFS**Mode 2 Depth (m) Mode 1 radians**streamwise**cross-stream**streamwise**cross-stream