Can G be Written as a Matrix of Matrices like H?

[EngWiPy]

Hello,

I have the following matrix of matrices

$$\mathbf{H}=\begin{array}{cc}\mathbf{A}&\mathbf{B}\\\mathbf{B}^H&\mathbf{A}\end{array}$$

where each element is a square matrix, A is a diagonal matrix of real numbers, whereas B is not (necessarily), and the superscript H means conjugate transpose.

Now I have the following matrix

$$\mathbf{G}=\mathbf{H}^H(\mathbf{H}\mathbf{H}^H+\alpha\mathbf{I})^{-1}$$

where 'alpha' and 'I' are a constant scalar and the identity matrix, respectively. Will this matrix exhibit the same structure as H. In other words, can we write G as:

$$\mathbf{G}=\begin{array}{cc}\mathbf{A}_G & \mathbf{B}_G \\ \mathbf{B}^H_G &\mathbf{A}_G \end{array}$$

Via simulation it looks like it does, but I am wondering how to prove this mathematically?

Thanks in advance

 

[CompuChip]

Maybe this will help:

Source: http://www.cs.nthu.edu.tw/~jang/book/addenda/matinv/matinv/