Universal optimality of the double-centred matrix under unitarily invariant norms for dissimilarity data

Let $D=(D_{ij}{)}_{i,j=1}^n$ be a symmetric dissimilarity matrix and let $D^{(2)}=(D_{ij}^2{)}_{i,j=1}^n$. We study the affine family of real symmetric matrices $$A(\mathbf{g})=\frac{1}{2}\text{\hspace{0.17em}⁣}(\mathbf{1}{\mathbf{g}}^{\mathrm{\top }}+\mathbf{g}{\mathbf{1}}^{\mathrm{\top }}-D^{(2)}),\mathbf{g}\in {\mathbb{R}}^n,$$ parametrised by a free diagonal vector $\mathbf{g}\in {\mathbb{R}}^n$, whose off-diagonal entries satisfy $$D_{ij}^2=a_{ii}+a_{jj}-2a_{ij},i\mathrm{\ne }j\mathrm{.}$$ We prove that the double-centred matrix $$A({\mathbf{g}}^F)=-\frac{1}{2}J{D}^{(2)}J,J=I-\frac{1}{n}{\mathbf{11}}^{\mathrm{\top }},$$ is the unique minimiser of the Frobenius norm over this family, with minimiser ${\mathbf{g}}^F$ given explicitly by $$g_k^F=\frac{1}{n}\sum _{i=1}^n{D}_{ik}^2-\frac{1}{2n^2}\sum _{i,j=1}^n{D}_{ij}^2,k=1,\dots ,n,$$ and that this same representative simultaneously minimises every unitarily invariant norm $$\underset{\mathbf{g}\in {\mathbb{R}}^n}{\min }\mid \text{\hspace{0.17em}⁣}\mid \text{\hspace{0.17em}⁣}\mid A(\mathbf{g})\mid \text{\hspace{0.17em}⁣}\mid \text{\hspace{0.17em}⁣}\mid ,$$ including the spectral norm, the nuclear norm, and all Schatten $p$-norms and Ky Fan $k$-norms. Thus the double-centred matrix, central to classical multidimensional scaling, admits a purely variational characterisation that does not depend on the choice of norm and requires no Euclidean realisability assumption on $D$.