# Properties of a matrix built via a "matricization" of a unit vector [closed]

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 $$4$$?

• Take a look at this. Feb 19, 2022 at 11:33
• 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. Feb 19, 2022 at 11:34
• @CarloBeenakker ... Didn't think about that... Thanks. Feb 19, 2022 at 11:37