See this book for Definitions and intro to their use in PCA:

https://www.amazon.com/Basics-Matrix-Algebra-Statistics-Chapman/dp/1498712363/ref=sr_1_1?keywords=nick+fieller+Matrices&qid=1574601072&s=books&sr=1-1

See this book for Full details on PCA:

https://www.amazon.com/Applied-Multivariate-Statistical-Analysi-Johnson/dp/9332549559/ref=pd_sbs_14_1/135-6522911-4180237?_encoding=UTF8&pd_rd_i=9332549559&pd_rd_r=3276f01a-db85-466c-ab0e-66fca0eec3b3&pd_rd_w=keHmb&pd_rd_wg=roNME&pf_rd_p=5873ae95-9063-4a23-9b7e-eafa738c2269&pf_rd_r=GSKXXF8CTD15KDSV5RXC&psc=1&refRID=GSKXXF8CTD15KDSV5RXC

Best wishes, David Booth

BTW you seem a bit confused on this issue.

From GUSTA ME website:

"As with other ordination techniques such as PCA and CA, PCoA produces a set of uncorrelated (orthogonal) axes to summarise the variability in the data set. Each axis has an eigenvalue whose magnitude indicates the amount of variation captured in that axis. The proportion of a given eigenvalue to the sum of all eigenvalues reveals the relative 'importance' of each axis. A successful PCoA will generate a few (2-3) axes with relatively large eigenvalues, capturing above 50% of the variation in the input data, with all other axes having small eigenvalues."

https://sites.google.com/site/mb3gustame/dissimilarity-based-methods/principal-coordinates-analysis

You might look into the broken stick model or the Kaiser–Guttman criterion for deciding on the number of axes. A P-value could be made... permutation... but not sure how it would be meaningful. Have a look here at related thread https://www.biostars.org/p/279919/.

The simple answer is that the eigenvalues are the amount of variance in each dimension. You can compute the proportion of variance in each dimension by dividing the eigenvalues by the sum of the eigenvalues.

Dear Aya Al-Busaidi

Your question:

what do eigenvalues on both the first and second components mean?

It is well known that the eigenvalue is a real or a complex number.

Before answering your question, can you tell me what do you mean by the first and second components?

Do you mean the real and imaginary parts of the complex eigenvalue?

Or the first and second components of the corresponding eigenvectors?

Best regards

Hello, Issam Kaddoura

I mean the numbers on both axis.

I mean the corresponding eigenvectors

Aya,

If what we have here is a PCA applied to distance measures, which is, in my experience, standard for the label PCoA, then as clearly specified by preceding replies, the eigenvalues have their usual meaning in respect to encompassing the input distance matrix.

In regard to p-values, I would encourage consideration of of the percentage of the sum of all eigenvalues attributable to your selected reduced dimensionality eigenvalues be used, rather than any constructed p-value.

You say, "first and second components" so it may be that you have already made up your mind about dimensionality. I suspect that such may come from your own expertise and that in the literature. I am quite sure that it will also be affected by the distance measures you use and how you may have normalized them. Were this an analysis of a covariance or correlation matrix, I would have suggested parallel analysis to help you in the dimensionality decision. As parallel analysis derives from simulation, it would be theoretically possible to taylor-make such for comparison of your distance eigenvalue decline with that expected with random eigenvalues, but I don't know of software to do that automatically -- others may. Again, this will depend on how you have normalized the distances.

In regard to Issam Kaddoura 's comment, if I understand the question, it would seem to me that the matrix that is analyzed is symmetric and full-rank, so I don't see how the eigenvalues would be complex, but again, I may be missing something.

My best on your research,

John