Equation 10.29 is not symmetric. The m12 value resulting from fitting Y2 to Yj differs from m21. If the matrices Yj and Y2 (with column vectors centred on their respective means) are transformed, before the analysis, to have unit sums-of-squares, the traces of the two cross-product matrices become 1. To accomplish this, compute the trace of (YjYj) and divide each value of Yj by the square root of the trace. The trace of (YjYj) , which is the same as the trace of (YjYj) , is easily computed as the sum of squares of all values in Yj. Do the same for Y2. For matrices Yj and Y2 transformed in this way, the two Procrustes statistics are now identical:
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