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My question is a bit related to this question: Inequality concerning inverses of positive definite matrices.
I am asking only out of curiosity, does the result hold true if the inequality means component-wise, assuming that the matrices are also invertible and elements are all non-negative? I tried with random matrices in Matlab, but so far, I am not getting any contradiction.
Any help will be really great.

Cherryblossoms
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1 Answers1

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The question is interesting but like this it depends on the sign of determinant as one can easily get a counter example $$A=\begin{pmatrix}1&2\\1&1\end{pmatrix}\overset{entry}{\le}B=\begin{pmatrix}2&2\\1&2\end{pmatrix}$$ $A^{-1}$ has a negative diagonal opposite to that of $B^{-1}$

For determinant equals one $$A=\begin{pmatrix}2&1\\1&1\end{pmatrix}\overset{entry}{\le}B=\begin{pmatrix}2&\sqrt{3}\\\sqrt{3}&2\end{pmatrix}$$

Toni Mhax
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  • But, can we say if determinant is positive in both the cases, then it holds? Or what possible conditions we might to put to get so. Just a hint would be enough. – Cherryblossoms Oct 10 '20 at 09:19
  • I just noticed that the matrices taken here are not symmetric. Can you please clarify if the example given is just to show that the result may or may not hold? Furthermore, what if the determinant also is positive. – Cherryblossoms Oct 19 '20 at 14:13
  • Yes it is easy to have counter examples – Toni Mhax Oct 19 '20 at 16:45