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Inverse of Matrix of Matrices

  1. Jul 25, 2013 #1
    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
     
    Last edited by a moderator: Jul 25, 2013
  2. jcsd
  3. Jul 25, 2013 #2

    CompuChip

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