Given any symmetric matrix $\mathbf{M} = \begin{pmatrix}
\mathbf{A} & \mathbf{B}\\
\mathbf{B}^\mathrm{T}& \mathbf{C}
\end{pmatrix}$, the following conditions are equivalent:
(1) $\mathbf{M}\succeq 0$ ($\mathbf{M}$ is positive semidefinite)
(2) $\mathbf{A}\succeq 0$, $\;\;(\mathbf{I}-\mathbf{A}\mathbf{A}^T )\mathbf{B}= 0$, $\;\;\mathbf{C} – \mathbf{B}^T\mathbf{A}^{\dagger }\mathbf{B} \succeq 0$.
(3) $\mathbf{C}\succeq 0$, $\;\;(\mathbf{I}-\mathbf{C}\mathbf{C}^T )\mathbf{B}= 0$, $\;\;\mathbf{A} – \mathbf{B}\mathbf{C}^{\dagger }\mathbf{B}^T \succeq 0$.
Then, if we are given that $\mathbf{A}\succeq 0$, $\mathbf{B}\succeq 0$ and $\mathbf{C}\succeq 0$, is there any way to verify (2) or (3) and therefore conclude that $\mathbf{M}$ will be positive semidefinite as well?
Best Answer
The conditions$\;\;(\mathbf{I}-\mathbf{A}\mathbf{A}^T )\mathbf{B}= 0$, $\;\;(\mathbf{I}-\mathbf{C}\mathbf{C}^T )\mathbf{B}= 0$, and $\mathbf{B} \succeq0$, do not follow from anything. W/out them the equivalence follows from the Sylvester criteria for positive definiteness and determinant formula for Schur complement. What is the source of the problem?