Consider two real symmetric positive matrices $A\geq0$ and $B\geq 0$ with the following block form
\begin{equation}
A=
\begin{pmatrix}\begin{array}{@{}c|c@{}}
A_{11} & A_{12} \\ \hline
A_{21} & A_{22} \\
\end{array}\end{pmatrix}, \quad B=\begin{pmatrix}\begin{array}{@{}c|c@{}}
B_{11} & B_{12} \\ \hline
B_{21} & B_{22} \\
\end{array}\end{pmatrix}.
\end{equation}
I need to know if $A_{12}B_{21}\geq 0$.
Thanks.
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tamih100
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Have you tried anything yet? – Suzu Hirose Jan 10 '23 at 06:01
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What do you mean by "positive" ? All entries positive or "symmetric definite positive" ? – Jean Marie Jan 10 '23 at 07:29
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Please answer our questions. – Jean Marie Jan 10 '23 at 08:24
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@JeanMarie sorry for the confusion. Is matrix $A_{12} B_{21}\geq 0$ in the sense of nonnegative eigenvalues? – tamih100 Jan 11 '23 at 16:35
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Having all eigenvalues $\ge 0$ is synonymous of a semi-definite positive matrix ; if it is $>0$ : it means "positive definite matrix". – Jean Marie Jan 11 '23 at 16:40
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@JeanMarie noted, thanks. – tamih100 Jan 11 '23 at 16:41
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The answer is no. Counter-example with the symmetric positive definite matrices : $$A=\begin{pmatrix}9&-3&-2&-1\ -3&7&-1&1\ -2&-1&7&-3\ -1&1&-3&3 \end{pmatrix} \ \text{and} \ B=\begin{pmatrix}4&1&-1&1\ 1&2&0&-2\ -1&0&4&0\ 1&-2&0&4 \end{pmatrix}$$. I let you compute $A_{12}B_{21}$. You will see that its eigenvalues are $-3,2$. In other cases one finds that $A_{12}B_{21}$ is non-symmetrical... – Jean Marie Jan 11 '23 at 17:29
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@JeanMarie Thanks – tamih100 Jan 15 '23 at 03:52