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look here my friend. How to prove the following equation? Or give a counter-example. Thank you so much

$$\text{rank}(A^+)\leq \text{rank}(A)$$

where $\text{rank}(A^+)$ represents the positive components of matrix $A$, e.g. $$\left[\begin{matrix}2 & -1.5 & 0\\-2.3 & 2 & 4.1\end{matrix}\right]^+=\left[\begin{matrix}2 & 0 & 0\\0 & 2 & 4.1\end{matrix}\right].$$

guorui
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1 Answers1

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It's not true. Try with $A = \begin{bmatrix}1& \color{red}{-1} \\ \color{red}{-1} & 1\end{bmatrix}$. Then $A^{+} = \begin{bmatrix}1& 0\\ 0& 1\end{bmatrix}$, which has greater rank than $A$.

Minus One-Twelfth
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