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Consider matrix subspace $\mathbb{C}^{m\times n}$. What is the minimum $d$ such that for any $d$-dimensional matrix subspace of $\mathbb{C}^{m\times n}$, there exists two distinct matrix $M,N$ such that $$ M^{+}M=N^{+}N $$ here $M^{+}$ denotes the hermitian conjugate of $M$.

The distinct means that there is no $\lambda$ such that $M=\lambda N$.

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    $\begingroup$ $d=1$. Take $M=-N\neq 0$. $\endgroup$
    – Fabian Wirth
    Commented Feb 22, 2023 at 17:47
  • $\begingroup$ Maybe you want $M^*M=N^*N$ but where there is no $\lambda\in\mathbb{C}^\times$ with $M=\lambda N$. $\endgroup$
    – Joseph Van Name
    Commented Feb 22, 2023 at 18:25
  • $\begingroup$ @Joseph Oh yes, thanks. $\endgroup$
    – gondolf
    Commented Feb 22, 2023 at 20:27

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