I need to prove, that the matrix $ \begin{pmatrix} A &B \\ B & C \end{pmatrix} $ has at least one positive eigenvalue, if known that $ A+4B+5C > 0 $.
I was told to show that $ \begin{pmatrix} 1 &2 \\ 2 & 5 \end{pmatrix} $ is positive definite. But I don't know what to do with that hint, and what is the connection between the inequality and and the matrix.
Two things I know and think that might be usefull is that the sum of eigenvalues is the trace of the matrix, and their product is the determinent
any usefull hints/directions?
big thank you