Slick correlation "plots" in Wikipedia entry for correlation. I just wonder if correlation should really be zero for the (3,2)-th case. I woukd say it should be slightly more than zero.
Actually, the (3,2) plot does have a correlation of zero, since it's just the (3,3) plot rotated. Here's a simple proof: let z = (x,y) and A be a rotation matrix (a 2x2 matrix with cos t on the diagonal and off-diagonal elements sin t and -sin t, where t is the angle of rotation). Then if V(z) = s^2 I (where I is the 2x2 identity matrix), then V(Az) = A V(z) A' = s^2 AA' = s^2 I since A is orthogonal.
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Actually, the (3,2) plot does have a correlation of zero, since it's just the (3,3) plot rotated. Here's a simple proof: let z = (x,y) and A be a rotation matrix (a 2x2 matrix with cos t on the diagonal and off-diagonal elements sin t and -sin t, where t is the angle of rotation). Then if V(z) = s^2 I (where I is the 2x2 identity matrix), then V(Az) = A V(z) A' = s^2 AA' = s^2 I since A is orthogonal.
Cool proof! Thanks!
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