It depends on what you mean by 'meaning'.

In both methods the size and direction come directly from the math, so have the same mathematical meaning. Thus variables pointing in the same direction and length by either method have the same mathematical properties,

However, how the factors are determined is different, with the result that the scientific meaning is not the same. Negation of a PCA is inconsequential (as long as the negation is consistently applied in decomposition and projection), negation of a LDA inverts the group prediction since decomposition is directed based on group membership.

If a PC flips sign then it self corrects for projection since the PC is flipped the scores flip and the data reconstructs with the same fidelity. What doesn't work is flipping the sign of scores and reconstructing with the unflipped PC.

LDA linearily combines its individual factors with a set of weightings to create a discriminant function - mess with signs and you change the outcome.

Dear James Renwick Beattie , thank you for your answer!!

I'll try to be more specific. I hope this images may help.

Look, the red cluster has the lowest value for 'dry.m.pre', as we can notice in the boxplot.

In the PCA, this cluster is on the opposite side of the 'dry.m.pre' arrow, in agreement with this. But in the LDA, the same arrow is pointing towards this cluster. That's the reason I doubt my interpretation of the LDA. The fact that the arrow is pointing to this group does not mean that it has higher values for that variable, right?

The only interpretation that makes sense to me is that this variable is the one that best defines this group. That doesn't mean that the values for this group are greater than for the others.

I'm using the standardize method of the decostand function (vegan package) to standardize the data, can this be a problem?

I hope I've made myself clear. Thank you!

LDA is a directed data reduction, so features point towards the group that they are frequent in and away from groups they are rare or low in. Remember that uniqueness can be defined by what you don't have as well as what you do have. So if all other groups have fairly consistent high levels of a parameter, and one group is uniquely low in that characteristic then a negative feature will appear in the discriminant function.

The arrows in the two chemometric techniques mean different in their formation and interpretation. The PCA loadings indicate the directions and strength of the variations of the selected parameters in the entire data set along PC1 and PC2 biplot , while in LDA the arrow drawn indicates maximum difference between the groups along the canonical 1 and canonical 2 with respect to the parameters under consideration.

PCA (Unsupervised pattern recognition technique) is a data reduction technique that tells you about the variations in the multivariable data set. The eigenvectors from the PCA tell you or give you idea on how a variable changes in value across your data set. It determines the degree or extent to which variables are related mathematically using covariance matrix. LDA (supervised pattern recognition technique) is used for statistical classification of samples. It determines the variables that distinguish one group from another by constructing discriminant function (DF) for each group. LDA establish a DF for each group given various independent variables and dependent variables.

Kindly refer to the links below for more guide.

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Best regards.