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I would like to define an inverse on matrices whose entries may be positive or negative infinity.

To formulate my problem precisely, suppose that I have a matrix $A$ and another matrix $B$. How do I easily compute $\lim_{x\to\infty} (A+xB)^{-1}$?

Of course this is not so interesting if $B$ is invertible. It is easy to see that, in this case, $\frac{1}{x}A + B$ is invertible for a suitably large $x$ and the inverse converge to that of $B$. So here the limit equals to $0$.

Another special case is if $B$ is diagonal with $k>0$ zero eigenvalues, ordered to the top. Here the limit equals to a 2x2 block matrix where the first block is the inverse of $A_{11}$, i.e. the corresponding block of $A$; and the other elements are all 0.

The problem is, how do I handle the special case? When is there a limit, and what does it equal to? Has this been investigated anywhere?

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  • $\begingroup$ Write $J=UBU^{-1}$ in Jordan normal form. Then your limit is $U^{-1}\left(\lim_{x\to\infty}(UAU^{-1}+xJ)^{-1}\right)U$. So if $B$ is diagonalizable, you're reduced to thediagonal case that you already analyzed. If $J$ is not diagonal, it's a bit more complicated,but should be computable. Presumably the eigenvalue zero blocks of $J$ are the ones that are important. $\endgroup$
    – Joe Silverman
    Commented Oct 16, 2015 at 20:02

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