So I am trying to determine if the difference of two matrices is positive(negative) semi-definite. The matrices, we will call them $\displaystyle W$ and $\displaystyle \Omega$, are partitioned as

$\displaystyle W=\begin{pmatrix}W_{11}&W_{12}\\W_{21}&W_{22}\end{ pmatrix}$

and

$\displaystyle \Omega=\begin{pmatrix}W_{11}&W_{12}-B\\W_{21}-B&W_{22}+B-B\cdot{W_{21}}-{W_{21}}\cdot{B}\end{pmatrix}.$.

My problem is that I wish to determine the definiteness of the matrix $\displaystyle A_{1}=\Omega-W$ or/and $\displaystyle A_{2}=W-\Omega$. These matrices are given by

$\displaystyle A_{1}=\begin{pmatrix}0&-B\\-B&B-B\cdot{W_{21}}-{W_{21}}\cdot{B}\end{pmatrix}$

and

$\displaystyle A_{1}=\begin{pmatrix}0&B\\B&-B+B\cdot{W_{21}}+{W_{21}}\cdot{B}\end{pmatrix}.$

We know from simple linear algebra that a matrix is positive(negative) semi-definite if and only if all of its principle minors are non-negative(non-positive). With this I have two questions: 1) Does the same hold true for block matrices? 2) If this does hold true the second principle minor, in block form, for both matrices is $\displaystyle -BB$, what does this mean exactly. Does this mean that the matrices are non-definite?

Thanks