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Hello,
I have the following matrix of matrices
[tex]\mathbf{H}=\begin{array}{cc}\mathbf{A}&\mathbf{B}\\\mathbf{B}^H&\mathbf{A}\end{array}[/tex]
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
[tex]\mathbf{G}=\mathbf{H}^H(\mathbf{H}\mathbf{H}^H+\alpha\mathbf{I})^{-1}[/tex]
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:
[tex]\mathbf{G}=\begin{array}{cc}\mathbf{A}_G & \mathbf{B}_G \\ \mathbf{B}^H_G &\mathbf{A}_G \end{array}[/tex]
Via simulation it looks like it does, but I am wondering how to prove this mathematically?
Thanks in advance
I have the following matrix of matrices
[tex]\mathbf{H}=\begin{array}{cc}\mathbf{A}&\mathbf{B}\\\mathbf{B}^H&\mathbf{A}\end{array}[/tex]
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
[tex]\mathbf{G}=\mathbf{H}^H(\mathbf{H}\mathbf{H}^H+\alpha\mathbf{I})^{-1}[/tex]
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:
[tex]\mathbf{G}=\begin{array}{cc}\mathbf{A}_G & \mathbf{B}_G \\ \mathbf{B}^H_G &\mathbf{A}_G \end{array}[/tex]
Via simulation it looks like it does, but I am wondering how to prove this mathematically?
Thanks in advance
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