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A rank inequality

Suppose $$M := \begin{bmatrix} M_{11} & \cdots &M_{1d} \\ \vdots & \ddots & \vdots \\ M_{d1} & \cdots & M_{dd} \end{bmatrix}$$ is a $d \times d$ block matrix such that $$M_{jk} = \sum_{i=1}^{r}(A_ …
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