= Centralizer of a Hermitian matrix in SU(n)
{title2=$S(U(n_1)\times\cdots\times U(n_k))$}
Let a <Hermitian matrix> have distinct real <eigenvalues> with multiplicities $n_1,\ldots,n_k$. A <unitary matrix> commuting with it must preserve each <eigenspace>: if $Av=\lambda v$, then $A(Uv)=UAv=\lambda Uv$. Conversely, any unitary transformation acting independently on these mutually orthogonal eigenspaces commutes with $A$. The <centralizer> in $U(n)$ is therefore $\prod_jU(n_j)$. Imposing total determinant one gives the displayed subgroup inside the <special unitary group> $SU(n)$.
For multiplicities $(2,1)$ in <SU(3)>, it is $S(U(2)\times U(1))\cong U(2)$, embedded by $U\mapsto\operatorname{diag}(U,(\det U)^{-1})$. Three distinct eigenvalues leave the diagonal <maximal torus> $U(1)^2$. This determines the unbroken group for a diagonal <adjoint scalar field> vacuum, including its global determinant constraint.
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