If the correlation-matrix, say R, is positive definite, then all entries on the diagonal of the cholesky-factor, say L, are non-zero (aka machine-epsilon). Btw, to use this tool for the collinearity-detection it must be implemented as to allow zero-eigenvalues, don't know, whether, for instance, you can use SPSS for this.
The number of onnon-zero entries in the diagonal indicate the actual rank of the correlation-matrix. And because of the triangular structure of the L-matrix the variables above the first occuring diagonal zero form a partial set of variables which is of reduced-rank. However, there may be some variables in that block, which do not belong to that set, so. So to find the crucial subset which contains only the multicollinearity you do several recomputations of the cholesky-decomposition, where you reorder the variables such that you find the smallest possible subset, which shows rank-reduction - so this is an iterative procedure.
(If needed, I'll show an example where I use my MatMate-program for the script, later).
As we see, that only 4 of 5 diagonal elements are non-zero (above machine-epsilon) we know, that the correlation matrix has rank 4 instead of 5 and we have collinearity. But we do not yet know, whether 4 variables are linnearlylinearly dependent or whether we have possibly a rank reduced subspace of even smaller dimension. So we try iteratively the rotation to triangularity, where the order of the variables $x_1$ to $x_5$ is systematically altered to identify any possible smallessmallest subset.
and we see,. that rank-reduction is already occuring if we ignore variable 3, - because the variables $x_1,x_2,x_4,x_5$ define already a 3-dimensional subspace (instead of a 4-dimensional one).
[24] l1=rotL1 = rot(L,"drei",5´4´1´2´3)
L1= f1 f2 f3 f4 f5
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x1 0.0952 -0.9492 0.3000 . .
x2 0.5355 0.8445 . . .
x3 -0.4407 -0.0397 0.7803 . 0.4421
x4 -0.5066 0.8622 . . .
x5 1.0000 . . . .
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Now we'vewe're nearly done: the subset of $x_2,x_4,x_5$ forms a reduced subspace and to see more, we put them at "the top" of the cholesky-process:
andWe see, that $x_1$ has a component outside of that reduced space, and $x_3$ has a further component outside of the rank 3 space, and are thus partly independent of that 2-dimensional subspace (which can thus be givegiven the term "co-planarity"). We can now decide, which of the three variables $x_2,x_4$ or $x_5$ can be removed to overcome the multi-collinearity problem.
[26] l1=choleskyL1 = cholesky(...) // something in your favorite software...
L1= f1 f2 f3 f4 f5
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x5 1.0000 . . . . Co-planar subset
x2 0.5355 0.8445 . . .
x4 -0.5066 0.8622 . . .
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x1 0.0952 -0.9492 . 0.3000 . further linearly independent
x3 -0.4407 -0.0397 . 0.7803 0.4421 variables
[update]: Note, that the candidates from which we would remove one, were not necessarily recognized by the inspection of correlations in the correlation-matrix. There the highest correlation is 0.8666 between $x_1$ and $x_4$ - but $x_1$ does not contribute to the rank-deficiency! Furthermore, the correlations between $x_2,x_4,x_5$ are all in an "acceptable" range when one wants to apply some jackknife-estimate for the removal of high-correlations assuming multicollinearity - one would not look at them as the most natural candidates from the set of bivariate correlations only.