Minimal spanning set ("conical basis") for 2x2 Hermitian PSD (positive semi-definite) cone?

Solution 1:

Consider the matrices $$ \pmatrix{ 1 & t \\ \bar t & t^2 } $$ for $t\in \mathbb C$. All are positive semi-definite. For $|t|>1$ they are no conical combination of your matrices: The entry $t$ fixes one of the matrices C-F, then there is no room to add positive multiples of A,B.

Afaik, this cone cannot be generated by finitely many matrices. So the result you are looking for might not be true.

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