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.