Is it possible that 3 vectors have all negative pairwise correlations?

Given three vectors a, b, and c, is it possible that correlations between a and b, a and c, and b and c are all negative? I.e. is this possible?

corr(a,b)<0corr(a,c)<0corr(b,c)<0

Answer

It is possible if the size of the vector is 3 or larger. For example

a=(1,1,1)b=(1,9,3)c=(2,3,1)

The correlations are
cor(a,b)=0.80...cor(a,c)=0.27...cor(b,c)=0.34...

We can prove that for vectors of size 2 this is not possible:
cor(a,b)<02(iaibi)(iai)(ibi)<02(a1b1+a2b2)(a1+a2)(b1b2)<02(a1b1+a2b2)(a1+a2)(b1b2)<02(a1b1+a2b2)a1b1+a1b2+a2b1+a2b2<0a1b1+a2b2a1b2+a2b1<0a1(b1b2)+a2(b2b1)<0(a1a2)(b1b2)<0

The formula makes sense: if a1 is larger than a2, b2 has to be larger than b1 to make the correlation negative.

Similarly for correlations between (a,c) and (b,c) we get

(a1a2)(c1c2)<0(b1b2)(c1c2)<0

Clearly, all of these three formulas can not hold at the same time.

Attribution
Source : Link , Question Author : Antti A , Answer Author : Community

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