In my matrix theory studies I recently came across the following problem:
We have the following real symmetric $ n \times n $ matrix $ A=\begin{pmatrix} d_1 & u^T E^\dagger v & u^T \\ v^T E^\dagger u & d_2 & v^T \\ u & v & E \end{pmatrix} $ where $ d_1,d_2 $ are real numbers and $E$ is a square $n-2\times n-2$ matrix, $u,v$ are column vectors of length $n-2$, T denotes transpose as usual and $ \dagger $ denotes generalized inverse (or classic inverse if it is easier to solve) and we know that any principal submatrix of $A$ that does not include entries $ a_{1,2},a_{2,1} $ is positive semidefinite, and we are asked to show $A$ is positive semidefinite.
I thought a direct approach might be difficult, so I though about using the characterization theorem of positive semidefinite matrices to show an easier equivalent condition. I would certainly appreciate all help.
Well, according to what you say if you choose
$$n = 4 $$ $$E = I$$ $$u = v = [1 \quad 1]^T$$ $$d_1 = d_2 = 1$$ You can show that matrix that the principle submatrices by deleting the second row and column are positive definite, however $A$ is not positive definite. The eigenvalues, in that case, are given as $\lambda = [-1,\ -0.2361,\ 1, \ 4.2361]$.
I hope i understood your question correctly.