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A block matrix $\mathbf{M}=\left[ \begin{array}{ccc} \mathbf{A} & \mathbf{B} \\ \mathbf{B}^T & \mathbf{C} \end{array} \right]$ is invertible if $\mathbf{A}$ and $(\mathbf{C}-\mathbf{B}^T\mathbf{A}^{-1}\mathbf{B})$ are both non-singular.

I am looking on the invertibility of the Schur Complement $(\mathbf{C}-\mathbf{B}^T\mathbf{A}^{-1}\mathbf{B})$ at the moment. Is there any known results on the existence of the inverse in terms of $\mathbf{A}$, $\mathbf{B}$, and $\mathbf{C}$? I'm assuming $\mathbf{A, C}$ are square (not equally but compatibly)

Cheers

Olumide
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Pioneer83
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  • What exactly do you mean by the "invertibility of the Schur Complement $(C ...)$"? You just made the assumption that is invertible beforehand... Is the binomial inverse theorem what are you trying to get? – usεr11852 May 07 '16 at 19:05
  • @usεr11852: I didn't. I was just giving an "if" statement. I was looking at the condition on A, B and C, that affect the existence of the inverse of the Schur Complement. – Pioneer83 May 09 '16 at 14:36
  • @usεr11852: I am aware of the matrix inversion lemma you're referencing, but again, it is still a function of the inverse a Schur complement, which I don't know if it would exist or not :( – Pioneer83 May 09 '16 at 14:38

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