I know that it is similar because of how my initial problem is posed. If you’re interested, see details below (if not, the end is marked):
The existence of such a B comes basically form the fact, that any hyperbolic polynomial f \in C[X,Y,Z] has a hermitian determinantal representation, i.e. one can write f as a determinant of some hermitian matrix: f = det(M(x,y,z)), where M is a hermitian 4x4 matrix of linear forms (homogenous polynomial of degree 1) in three variables x, y, z. M is basically something like xA + yB + zC with some hermitian 4x4 matrices A, B, C over C and such that f = det(M(x, y, z)).
What I did was to compute the first column of the adjuct matrix of M (it’s easier) and then out of the identity M \cdot M^adj = det(M) \cdot I = [f;0;0;0] by coefficient comparision (for that I solved some systems of linear equations Atilde x = b and computed the kernel of the representing matrix Atilde) I ended up with this 4 dimensional vector space V which in a way is just A, B and C stacked upon each other like so: [A;B;C]. And because of the existence of the hermitian determinantal representation (which also satisfies the Atilde x=b from above) I basically know that what I’m looking for must live in V too, i.e. I need some appropriate change of basis of V, so that then [A;B;C] are hermitian. So I’m trying to figure out how to do that. I was thinking about splitting V in A, B and C, force them to be hermitian and figure out how to glue them back together (and if that deglue-ing destroys things at all, I’m not sure, but I suppose it can happen) and then be done, so I’m exploring possibilities.
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Thank you for bringing me to eigen, this might help! I’m aware that the choice of B and P is not unique, but thanks for pointing that out. I also can’t make something out of my theory to help me find a suitable basis, because the actual statement is not directly connected to this.