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Having square matrices $M_j\in\mathbb{R}^{m\times m}$, with $j\in\{1\,..\,n\}$, what notation should be used to express the following sequence of multiplications?: \begin{align} M_n\to M_{n-1}M_n\to M_{n-2}M_{n-1}M_n \to \ldots \to M_1\ldots M_{n-2}M_{n-1}M_n \end{align}

I've thought of using the productory: $\prod^{1}_{j=n}M_j$, with an additional disclaimer indicating that the operations are left-multiplications. However I feel there may be a better way since I believe the standard use of $\prod$ is for right-multiplications.

The closest questions that I've found are this one and this one. However, left-multiplications are not contemplated.

Any help is much appreciated.

gtg
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    I would avoid any fancy notation as different readers may have different ideas of what order the product is taken. I'd just say: consider, for each $k$, the matrix $P_k=M_1M_2\ldots M_k$. – Ruy May 23 '22 at 22:14
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    @Ruy I see your point. In fact I may have been complicating myself. $$P_k = \prod_{i=k}^n M_i$$ works best as suggested by ecrin. – gtg May 23 '22 at 22:18

2 Answers2

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If you want the sequence $N_k=M_kM_{k+1}\dots M_{n-1}M_n$, it suffices to write $$N_k=\prod_{i=k}^{n}M_i.$$

If you want to write the $n$-th term, $i.e.$ $M_1M_{2}\dots M_{n-1}M_n$, remark that the product is associative, that is it doesn't change if you consider $AB$ as $B$ right multiplied by $A$ or $A$ left-multiplied by $B$, so I'd suggest the usual notation $$\prod_{i=1}^{n}M_{i}.$$

Finally, if you want to write $M_nM_{n-1}\dots M_{2}M_1$, as in the first linked solution, the best option could be $$\prod_{i=1}^{n}M_{n-i+1}.$$

RobPratt
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ecrin
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Probably this one $$\prod_{j=n-k} ^ {n} M_{j}$$

kolobokish
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