Varying lengths of eigenvectors on a PCA biplot [duplicate]

Question

I'm conducting a PCA in Matlab on standardized variables. My goal is to interpret

• angles = loadings, correlations bw. variables and PC-axis
• directions = vectors point to the direction of the variable
• length = ?

of the eigenvectors from the biplot.

In my biplot I get eigenvectors of different length. Could you please help me to understand why is it the case?

Also, I'm confused what does the length of the eigenvector mean if my data set is standardized? I summarize my questions/sources of my confusion in the two following questions:

1. First: I red that while conducting PCA on not standardized data (using covariance matrix) the length indicates the standard deviation of the variable. So what do different lengths of my eigenvectors mean while using correlation matrix? I found some answers online like:

   The length of a variable vector in the ordination plot reflects its contribution to the ordination. That is, variables with vectors which appear longer than others in a given ordination were more important in building the PCs used in that ordination.

   But I thought that the importance of variables w.r.t. ordinations is already reflected in the loadings (in the angles bw. the variables and PC-axis but not in their lengths). Could you please give me a starting point from which I can elaborate this matter further?

2. Second: shouldn't all eigenvectors be of equal length (=1) anyways as it is imposed by construction?