# Matrix Inversion Lemma for Infinite Matrices

Assume all matrices are real. Suppose $A$ is a positive definite matrix of size $n \times n$, while $H$ is a $\infty \times n$ matrix and $D$ is an infinite matrix with a diagonal structure, that is only nonzeros on the diagonals, i.e. size $\infty \times \infty$. I would like to find the inverse $(A + H^{T}DH)^{-1}$. Can the matrix inversion lemma be applied in this case, or is the matrix inversion lemma only limited to finite matrices? If the lemma does not apply, what alternative method is required to find the inverse analytically? Thanks all.

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What do you call matrix inversion lemma ? – Denis Serre Aug 29 '12 at 13:24
Woodbury formula, or the matrix inversion lemma is found here en.wikipedia.org/wiki/Woodbury_formula – ptuls Aug 31 '12 at 1:50

Hmm, if all dotproducts in $X = H^T D H$ are convergent, then I don't see any special problem: the $X$ matrix simply has finite rank, and you do not encounter any specific problems due to this configuration. However, the rank of $(A+X)$ might be reduced - but this is a general problem for any $(A+X)^{-1}$-problem...