I am considering $$ \min_{M\in \mathcal{M}} \|X - M\|:=x \neq 0, $$ where $X$, $M$ are $m\times n$ matrices, $\|\cdot\|$ is spectral norm and $\mathcal{M}$ is a matrix subspace. I wonder to what extent the solutions are unique, i.e. what can we say about the set $$ \mathcal{M}^* = \{M|\|X - M\|=x\}. $$ Currently, I only know it is a convex set. For $M_1, M_2 \in \mathcal{M}^*$, let $M = p M_1 + (1-p) M_2$, $p \in (0,1)$. For any vector $v$, using Cauchy's inequality, we have $$ v^T(X - M)^T(X - M)v \leq x^2, $$ thus $M \in \mathcal{M}^*$. Moreover, the equality holds only when $$ (X - M_1)v = (X - M_2)v = u,~~~~s.t.~u^T u = x^2. $$ I wonder whether there are more properties I could say about this set, e.g. its dimension/the properties of the common eigenvectors/the condition under which the solution is unique.
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