  We prove failure of the natural formulation of a matrix weighted bilinear
Carleson embedding theorem, featuring a matrix valued Carleson sequence as well
as products of norms for the embedding. We show that assuming an A2 weight is
also not sufficient. Indeed, a uniform bound on the conditioning number of the
matrix weight is necessary and sufficient to get the bilinear embedding. We
show that any improvement of a recent matrix weighted bilinear embedding,
featuring a scalar Carleson sequence and inner products instead of norms must
fail. In particular, replacing the scalar sequence by a matrix sequence results
in failure even when maintaining the formulation using inner products. Any
formulation using norms, even in the presence of a scalar Carleson sequence
must fail. As a positive result, we prove the so called matrix weighted
redundancy condition in full generality.
