Given scalars $u,v$ s.t. $0<u,v<1$, we seek the properties of the matrix defined by
$$P=A+uB+vC+uvD+u^2E+v^2F$$
A is symmetric and positive definite. $B,C,D,E,F$ are symmetric, but might not be positive definite. The matrices are of arbitrary size.
What can we say about $P$? If we assume all matrix inverses exist, what can you say about $P$? What other assumptions reveal information about $P$?
Update: We know that we can rewrite $P$:
$$ P = (I,uI,vI)\left(\begin{array}{ccc}A&B/2&C/2 \\ B/2& E & D/2 \\ C/2&D/2&F \\ \end{array}\right)\left(\begin{array}{c}I \\ uI\\vI \end{array}\right)$$
Identities that might be relevant https://en.wikipedia.org/wiki/Bunch%E2%80%93Nielsen%E2%80%93Sorensen_formula https://en.wikipedia.org/wiki/Binomial_inverse_theorem https://en.wikipedia.org/wiki/Sherman%E2%80%93Morrison_formula
This is a problem I ran into while trying to prove something about my research. Any ideas would help!