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Question: Show that if $A \in C^{n \times n}$ , $ A \ge 0 $ and A is singular, then there exists a sequence of matrices $C_k$, $k = 1,2,...$ such that $C_k \ge 0$, det $C_k = 1$ and trace $AC_k \le 1/k$

My approach: I have no idea how to even approach this exercise, any help is highly appreciated.

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Hint. If $A=B\oplus0_{(n-r)\times(n-r)}$ where $B>0$ and $r=\operatorname{rank}(A)$, consider $C=\left(\varepsilon I_r\right)\oplus\left(\varepsilon^{-r/(n-r)}I_{n-r}\right)$ for some positive scalar $\varepsilon$. What are $\det(C)$ and $\operatorname{trace}(AC)$ in this case?

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  • $\begingroup$ Hi, thanks for the hint. I'm still not sure how to find det($C$) and trace($AC$). $\endgroup$ – Ilan Aizelman WS Apr 23 at 14:00
  • $\begingroup$ I do. But I don't understand how $C$ is constructed. Same about $A$. $\endgroup$ – Ilan Aizelman WS Apr 23 at 17:22
  • $\begingroup$ @user1551 nice hint, $det(C)$ is indeed 1, but how do you calculate trace of $AC$ ,you dont know what is trace of $B$ after all... also how do you construct a sequence from that $C$? $\endgroup$ – user2323232 Apr 23 at 18:45
  • $\begingroup$ @user2323232 The trace of $B$ is precisely the trace of $A$. $\endgroup$ – user1551 Apr 23 at 19:18
  • $\begingroup$ @IlanAizelmanWS Do you know what is the direct sum of two matrices? $\endgroup$ – user1551 Apr 23 at 19:18

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