Suppose I have a unit vector $\vec v$, and I write it as a matrix, e.g., $16$-vector $\vec v=(v_1,\dots,v_{16})$, where $v_i$ is the $i$-th entry of the vector $\vec v$, is written as follows

$$\begin{bmatrix} v_{1}& v_{2}& v_{3}& v_{4}\\ v_{5}& v_{6}& v_{7}& v_{8}\\ v_{9}& v_{10}& v_{11}& v_{12}\\ v_{13}& v_{14}& v_{15}& v_{16} \end{bmatrix}$$

Are there some properties of this kind of matrices (some references might be okay), especially properties about the singular values of this matrix (e.g., the singular values are less than $1$ and the sum of the singular values are less than $2$, which are found numerically) when the dimension is not bigger than $4$?

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    $\begingroup$ Take a look at this. $\endgroup$ Feb 19, 2022 at 11:33
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    $\begingroup$ the only constraint your matrix $M$ has is that ${\rm tr}\, MM^\top=1$; so the sum of the singular values squared equals 1; that's all. $\endgroup$ Feb 19, 2022 at 11:34
  • $\begingroup$ @CarloBeenakker ... Didn't think about that... Thanks. $\endgroup$
    – narip
    Feb 19, 2022 at 11:37