I have exactly the same question as the person who is asking in this link:
https://stackoverflow.com/questions/59780402/in-the-r-package-geomorph-how-do-i-interpret-the-matrix-of-pc-loadings-as-retur
All I know is that the number of rows and columns in this matrix are double the number of landmarks because one of them (either rows or columns) represents the variables, that is, the x- and y- coordinates of the landmarks within the tps format. The main issue with interpreting this matrix would be figuring out how does R arrange the variables. I'm guessing this would be 1x, 1y, 2x, 2y, 3x, 3y,...
Also this does not explain why columns and rows are BOTH double the number of landmarks. I'm (again) guessing that columns are (maybe) representing variables, and rows are (maybe) representing a number of principal components arranged by variation percentage explained, set by default to be the same as variables.
I would not try to interpret the pattern of loadings by reading the numbers. Instead, I would plot the results. The function returns the shapes at the extreme values for each PC, so you can use the plotRefToTarget function to show the difference between the two extremes (or you could show the change from the mean to an extreme. In this example, the shape data are coordinates of landmarks, in a 3D array, in a file called "niger.shape". This returns a plot of the scores and the changes along the first two PCs, from the shape with the minimum to the maximum score along each.
niger.gm
Plotting the data is a good idea but interpreting the data from the Eigenvectors (PCs) below is quite easy too. The output here shows the first four of the possible 26 eigenvectors (the columns) and probably account for a great deal of the variance in the data.The rows are the coefficient of the original landmark variables that rotates them in 26 dimensional space that makes the data orthogonal (uncorrelated). You should have some input that describes the eigenvalues which tell you just how much variance each of these uncorrelated eigenvectors (PCs) explains.
The first eigenvector, by definition, describes the largest component of variation among the 26 (the eigenvalues are the variances explained by each PC).
The most influential components of this first PC are those with the largest absolute values as they determine the angle of the vector in multidimensional space. The signs show the direction of the vectors. So the coefficient for landmark [5,] (0.548) is the most influential variable in the first PC and it is contrasted with the next largest value [14,] (-0.401). So these two large landmarks play a large role in explaining the variation among all 26 rows and they are inversely correlated with one another. Other largish contributions come from [16,] [1,] [2,] and a few others. the negative values are inversely related to the positive values of the gm2$rotation.
Each of the other PCs are interpreted in the same manner so that the most important determinant of the direction of the second PC is [6,]. The length of the second PC is somewhat shorter than that of the 1st PC and so describes somewhat less of the total variance among the 26 original variables.
So these vectors tell you which landmarks explain the most variation among the taxa you are measuring. A really nice analysis would concentrate 90% (say) of the variance in the first three columns (PCs) thereby reducing the 26 dimensional of the original data into 3 dimensional space that contain most of the interesting and most influential landmarks as the largest coefficients.
If the first four or so PCs explain most of the data then you have a smaller, more interpretable matrix than the original matrix for describing how each taxa is related by the various landmarks. If on the other extreme, the 26 eigenvalues all are about the same size that the PC rotation has not resulted in reducing the dimensionality of the original matrix.
[,1] [,2] [,3] [,4]
[1,] -0.292637442 -0.24781965 -0.106858931 0.01109767
[2,] 0.308917172 -0.393705793 -0.059492315 -0.07341983
[3,] 0.030000517 -0.081826909 -0.065424375 0.09800728
[4,] 0.052069888 -0.165765161 -0.182545759 -0.01158679
[5,] 0.548194257 -0.381569139 -0.018172830 -0.01715879
[6,] -0.006640823 0.509101670 -0.110753022 0.07658526
[7,] -0.010698931 -0.016356517 -0.003072061 0.04752315
[8,] 0.055072685 0.043019180 -0.011659494 0.10461193
[9,] -0.104866710 -0.047445617 -0.016123797 -0.06791883
[10,] -0.022524580 -0.050072314 0.103334864 0.16119988
[11,] -0.229010887 0.016002306 0.348741743 -0.15581785
[12,] -0.037487586 -0.124647558 0.147662540 0.07513417
[13,] 0.181659233 0.257690024 -0.159775115 -0.34637649
[14,] -0.401315680 -0.177654634 -0.267710400 -0.13799427
[15,] 0.120250466 0.287037481 -0.160233065 -0.23500784
[16,] -0.308168182 -0.098020742 -0.316002315 -0.11907409
[17,] -0.201954619 0.002635231 0.103542233 -0.06862048
[18,] 0.014172165 -0.003263227 0.167454235 0.22778472
[19,] -0.139870351 -0.038388240 -0.043993696 -0.03608711
[20,] -0.060611865 0.111322129 0.389121041 0.18020552
[21,] 0.084694959 0.083795575 0.048581716 0.40593457
[22,] 0.084609928 0.040567815 0.521324332 -0.52082230
[23,] 0.025200160 0.191674102 -0.008007330 0.34023543
[24,] 0.237283438 0.264741813 -0.268730192 -0.14407308
[25,] -0.010960652 -0.025428638 0.080795508 0.02418930
[26,] 0.084623439 0.044376822 -0.112003514 0.
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