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If we have a sum of products of two conformable matrices: $\sum_{i=1}^NA_i'B_i$, I would like to understand if the following is generally true:

$$ \left\|\sum_{i=1}^NA_i'B_i \right\|\leq \max_{1\leq i\leq N} \left\| A_i\right\|\left\|\sum_{i=1}^NB_i \right\| $$

I do not require that $A_i$ or $B_i$ (or both) are positive definite for all $i$.

Ovi
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  • see first answer in https://math.stackexchange.com/questions/1393301/frobenius-norm-of-product-of-matrix – Oria Gruber May 09 '22 at 09:58
  • Thanks a lot! However, I see a discussion on submultiplicativity. I would for sure answer my questions if I had pushed the norm inside the sum. Yet, the norm is of the total sum of the second matrix. – Ovi May 09 '22 at 10:19

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