# Product of random matrices which commute nearly certainly Answer

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Product of random matrices which commute nearly certainly

In the paper “Matrix concentration for products” it’s said, that the next is simple to display.
Let $$X_1,dots X_n$$ breathe unbiased, bounded, sq. matrices, which commute nearly certainly. Define $$Y_i=I+frac{X_i}n$$. Then
$$log mathbb{E}|Y_ndots Y_1|leq frac 1 n |sum_{i=1}^nmathbb{E}X_i| +Oleft(sqrt{frac{log d}n}privilege)$$
$$|cdot|$$ is the spectral norm and $$d$$ is the dimension of our matrices.
I do know the weaker inequality for matrices which don’t essentially commute nearly certainly with $$sum_{i=1}^n|mathbb{E}X_i|$$ as a substitute of $$|sum_{i=1}^nmathbb{E}X_i|$$.
Is this actually simple to display? How do I show it?

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