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Let $A \in \mathbb{R}^{n \times n}$ be a symmetric positive definite matrix whose non-diagonal elements are all non-positive, determine whether all the non-diagonal elements of $A^{-1}$ are non-negative or not.

This is one problem of my final exam on linear algebra, and I didn't solve it during the examination.

I verified the cases of $n \leq 3$ and guess that this should be correct, but I don't know how to deal with non-diagonal elements of a matrix properly.

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  • $\begingroup$ Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be closed. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers. $\endgroup$
    – Arthur
    Jan 27, 2020 at 20:39
  • $\begingroup$ One way to see it is the following: Observe that in the Cholesky decomposition $A=LL^*$, the lower triangular $L$ has positive elements in the diagonal and all elements under the diagonal are non-positive. Apply induction and the recurrences in the link. Now, $A^{-1}=(L^*)^{-1}L^{-1}=(L^{-1})^*L^{-1}$. Observe that $L^{-1}$ has all entries non-negative ... $\endgroup$
    – OscarRascal
    Jan 27, 2020 at 21:07
  • $\begingroup$ ... This you can see, by induction, from the recurrences for Forward substitution, when you express $x_i$ in terms of the $b_1,b_2,...,b_n$ only. Therefore, $L^{-1}$ and $(L^{-1})^*$ have all their entries non-negative. Hence $A^{-1}=(L^{-1})^*L^{-1}$ has all its entries non-negative. $\endgroup$
    – OscarRascal
    Jan 27, 2020 at 21:07
  • $\begingroup$ If you have heard about M-matrices or monotone matrices in the course you may know that they are inverse positive. $\endgroup$
    – A.Γ.
    Jan 28, 2020 at 15:30

1 Answer 1

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Hints. Let the diagonal part and off-diagonal part of $A$ be $D$ and $-F$ respectively. Let $X= D^{-1/2}FD^{-1/2}$. Justify that $X\ge0$ entrywise, $\|X\|_2<1$ and $$ A^{-1}=D^{-1/2}(I-X)^{-1}D^{-1/2}=D^{-1/2}(I+X+X^2+\cdots)D^{-1/2}. $$ The rest should be easy.

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  • $\begingroup$ This kind of decomposition is extremely common for e.g. the graph Laplacian. OP's matrix is essentially a non-singular Laplacian. Not sure how I missed this. $\endgroup$
    – user8675309
    Jan 28, 2020 at 9:21

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