Least squares estimation with quadratic constraints (M*M = 0)

[Michael02]

Hello there,

currently I am trying to solve a least squares problem of the following form:

$min_{M}$ ||Y - M*X||$^2$

where M is a 3x3 matrix and Y and X are 3xN matrices. However, the matrix M is of a special form. It is a rank 1 matrix which satisfies M*M = 0$_{3x3}$ and the trace of M is zero, too.

Enforcing that the trace is zero seems rather easy, since it is a linear constraint. But I have no idea how to enforce M*M = 0$_{3x3}$, as it includes multiple quadratic constraints.

Does anyone have an idea how to solve this problem?

 

[fresh_42]

If $M$ is nilpotent of degree $2$ and of rank $1$, then there should be a basis, in which $M$ takes the form $M=E_{13}$, i.e. a $1$ at position $(1,3)$ and $0$ elsewhere.