I am trying to characterize matrices with a certain property :
Define $U$ as an $n \times n$ matrix (over C or R; you can also assume
that it is unitary or orthogonal if it helps). Now take $n$
unknowns and transform by $U$ : $y=Ux$; I want the product
$y\_1 y\_2 \cdots y\_n = \sum c\_i x\_i^n$; so basically I don't
want any "cross terms". In the case of $n=2$ this would put
a restriction on $U=((a,b),(c,d))$ : $ad+bc=0$; and $y\_1y\_2$ would be $c\_1x\_1^2+c\_2x\_2^2$; My question
is : are such matrices known in the literature by some name?
and can they be characterized in any nice way? (minimal number
of parameters,...)



characterize certain type of matrices
I'm guessing that these don't have a name, as they are fairly degenerate matrices.


If more than two of the $c\_i$ are nonzero, then no matrices satisfy these criteria. This follows since the Fermat curve
$$X^n + Y^n = Z^n$$
is irreducible and a decomposition as you describe would make it a union of lines.


If two of the $c\_i$ are nonzero, then all such matrices of this form have two columns $j, j'$ with nonzero entries and have $M\_{ij}/M\_{ij'}$ going through all $n^{th}$ roots of some nonzero complex number.


If one $c\_i$ is nonzero, then only one column has nonzero entries, and all its entries are nonzero.


If all the $c\_i$ are zero, then some row of $M$ is zero.


