If the two matrices differ in K 1 rows and K 2 columns, and ({lim }_{N to infty } frac {K_{1}+K_{2}}{N} = 0), then A N ∼B N. Denote the (m,n -th elements of A N and B N as a mn and b mn,n -thelements.
In this figure, row A shows that S3 is the only non-zero component of S1 when the matrices differ in shape but not in orientation, and row B that S2 is the only non-zero component when the matrices differ in orientation but not in shape.
First, I studied the procedures' power and Type I error through computer simulations that considered covariance matrices differing in shape, orientation or both, in different number of variables and sample size situations.
Four kinds of sample matrix comparisons were made: between samples taken at random from the same population, between samples from populations whose covariance matrices differed in orientation, whose matrices differed in shape, and whose matrices differed in both orientation and shape.
The phylogenies inferred from the three data matrices differed in tree shape and support for specific relationships (Figs 1, 2, 3).
It is not possible to directly assess the incongruence between the topologies from the concatenated cpDNA dataset and the ITS locus because the data matrices differed in the number of individuals.
In these simulations, CPCA should declare the matrices as "unrelated" when they differed in orientation, because in that case they would not share any eigenvector in common and as "CPC", i.e., sharing all eigenvectors but being not proportional when the matrices differed in shape.
Since the adhesion in nanotube-based composite is universally present in the form of van der Waals (vdW) interaction, the shear stress and the axial displacement of nanotube and matrix differ in the interface zone and are not the same.
When matrices differ both in shape and orientation, there is an intermediate angle that minimizes their differences in total variance explained.
Ag NPs synthesized in polyelectrolyte matrices differ from those prepared in non-ionic branched or linear matrices described previously [28, 29].
This is problematic in practical terms, since we often wish to compare nestedness of matrices that differ in these basic properties; in fact, cases where we compare empirically derived matrices with identical size and fill are an exception.
