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If A, B, C and D are four variables, the correlation matrix will be 4x4 matrix with elements,

$$ \left[ \begin{matrix} 1 & Corr(A,B) & Corr(A,C) & Corr(A,D) \\ Corr(B,A) & 1 & Corr(B,C) & Corr(B,D) \\ Corr(C,A) & Corr(C,B) & 1 & Corr(C,D) \\ Corr(D,A) & Corr(D,B) & Corr(D,C) & 1 \end{matrix}\right] $$

I have several questions: 1) What is the maximum possible value for the correlation terms in the matrix. I mean in 2D the maximum value of correlation is 1 but is it true in 4D or in general in nD?(Since the variables are dependent on each other, I assume this correlation will have to obey some relation).

2) In 3D if I know the $Corr(A,B)$ and $Corr(B,C)$, I can calculate the $Corr(A,C)$ using $$a \ge bc-(1-b^2)^1/2(1-c^2)^1/2$$ But in 4x4 matrix case if I have $Corr(A,B)=0.5774$ and $Corr(C,D)=-0.378$, is it possible to find the maximum value or bound of the remaining correlation terms?

Please Help

Update Thank you all for your answers. Actually I know that in 3D if r1 = r2 = r3 = r then it will be in the range $[-1/2,1]$, but if I have 4 random measurements, then what will be its range under equality. Furthermore, if I have given bound for the correlation of two variables, can I get tight bound for the other variables?

Suppose I have $A = [6$ $ 0;0$ $ 2]$ and $B = [2$ $ -1;-1$ $ 3]$ plotted as:enter image description here Now, if the within-set A and B covariance (correlation) is constant, what is the maximum bound of the between-set (covariance) correlation(if A & B is constant what is the bound of X in $[A$ $ X;X $ $B]$)? I hope, this will clear the question now!

tehseen fatima
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    The question about correlation between 3 random variables has been answered repeatedly on this site. See, for example, [this question](http://stats.stackexchange.com/q/72790/6633) which has a masterly answer by @whuber and note the _numerous_ `Related` questions that show in the right-hand column of the page. For four or more variables, the calculations are messier. – Dilip Sarwate Dec 04 '15 at 13:48
  • Yes, it was answered very clearly for 3D but I didn't understand its extension in 4 or higher dimensions. So I posted it here to get a clear answer. – tehseen fatima Dec 05 '15 at 06:15
  • If you would only **read** the answer by whuber that I referred to, you will discover that he points out that the _average_ correlation for $n$ variables is bounded between $-\frac{1}{n-1}$ and $1$, and that if all the correlations have the same value $\rho \in [-\frac{1}{n-1},1]$, then there exist random variables (even multivariate normal random variables if you like) with that correlation matrix. This answers your _new_ question _"but if I have 4 random measurements, then what will be its range under equality."_ – Dilip Sarwate Dec 05 '15 at 15:15

3 Answers3

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1) It is possible for all 4 variables to have correlation of 1 with all other variables. This would happen if all 4 variables are either the same or are a linear multiple of one another

2) As long as your correlation matrix is positive definite, it is valid. (It also must be symmetric and have 1s along the diagonal and each entry must be between 1 and -1).

Check this out: http://www.theactuary.com/features/2013/09/modelling-mastering-the-correlation-matrix/

Chechy Levas
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The average value of the $6$ correlations must be at least $-\frac 13$. More generally, for $n$ variables, the lower bound is $-\frac{1}{n-1}$; see this answer for a short proof. So, as pointed out by userE, it is entirely possible that one of the unknown correlations has value $-1$, or, for that matter, $+1$. Since you are seeking an upper bound on the maximum, you will be disappointed by the answer: the upper bound is $+1$ (which you knew even before you thought of $4$ random variables) and cannot be improved upon.

Community
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Dilip Sarwate
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But in 4x4 matrix case if I have Corr(A,B)=0.5774 and Corr(C,D)=−0.378, is it possible to find the maximum value or bound of the remaining correlation terms?

No, this is not possible, because you have no information on how the variables A or B are related to C or D. Their relationship could be anything from no correlation at all, to perfect positive or negative correlation.

userE
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