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

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SUMMARY

The discussion centers on the mathematical structure of the matrix G, derived from the matrix H, defined as \(\mathbf{H}=\begin{array}{cc}\mathbf{A}&\mathbf{B}\\\mathbf{B}^H&\mathbf{A}\end{array}\). The matrix G is expressed as \(\mathbf{G}=\mathbf{H}^H(\mathbf{H}\mathbf{H}^H+\alpha\mathbf{I})^{-1}\), where A is a diagonal matrix and B is not necessarily diagonal. The user seeks to determine if G can also be represented in a similar matrix of matrices form, specifically as \(\mathbf{G}=\begin{array}{cc}\mathbf{A}_G & \mathbf{B}_G \\ \mathbf{B}^H_G &\mathbf{A}_G \end{array}\). Initial simulations suggest that G retains the structure of H, prompting a request for mathematical proof.

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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
 
Last edited by a moderator:
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Maybe this will help:
img11.gif


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

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