There is a very simple argument that shows that MDS and PCA achieve the same results.

This argument has 2 important components. The first of these shows that an eigendecomposition of a gram matrix can be used for dimension-reduction.

PCA leverages the singular value decomposition (SVD). Given a matrix $ X $, the SVD is $ X = USV^{T} $.

We drop columns from X by using $ US_{t} $ where we drop some rows and columns from S.

This is also conviently obtained using an eigendecomposition of the covariance matrix.

Working with the gram matrix, we have $ XX^{T} $ and when expressed in terms of $ U $, $ S $ and $ V $, we have $ XX^{T} $ = $ (USV^{T})(VSU^{T}) $.

Simple algebra tells us that this is equal to $ US^{2}U^{T} $. The spectral theorem tells us that this the eigendecomposition of the gram matrix will return this decomposition. $ U $ and $ S $ can be retrieved and a dataset with fewer dimensions can be obtained.

The second part of the argument involves proving that a matrix of distances is indeed a gram matrix. This argument was discussed in a previous post.