Are there conditions (constraints) that the elements of the correlation matrix must meet? (For example, the linear correlation coefficient of X1 and X2 is 0.99, and it is the same between X2 and X3. Then between X1 and X3, it can be any low, or must it be greater than a given value?) If there are such constraints, what are they?
d1, d2,... dn be a set of vectors in R^n which are linearly independent, hence, the Matrix D=[ d1 d2... dn] is full rank (invertible), hence the associated covariance matrix H= D'*D is symmetric, positive definite.
But linear independence can be associated with poor mutual orthogonality as well , in the best case these vectors are mutually orthogonal and H is diagonal.
In a general case, these vectors would continue to span and form basis for R^n but their relative orientations may be such that it is far from the case of mutual orthogonality: as an example of this think to two linearly independent vectors and a third one marginally lying outside of the plane containing the other two: this situation and the more generalized cases of this, is quantified by estimate of " multicollinearity".
If the vectors d1, d2,...., dn are such that they form a strongly multicollinear set, the correlation between a pair of vectors of this set can be very high, and positive or negative, that is close to magnitude 1.
Correlation matrix is a measure of this orientational structure associated with these set of vectors d1,.... dn, if the vectors are mutually orthogonal, all the off diagonal terms in the correlation matrix would be zero.... in case of strong multicollinearity, the correlation matrix would be still full rank, but close to singularity.
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