# Matrix product notation

My lecturer has used some notation that I've never seen before: it is a (matrix) product symbol with a left-to-right arrow over the top. Does anybody know what this means?

Edit: It looks like this:

• Perhaps you could upload it as an image... to see what it actually looks like – Tolaso Aug 10 '15 at 11:56
• I think the simplest answer is to ask your lecturer what he/she means by it. – wltrup Aug 10 '15 at 12:21
• @Tolaso not statistics. It was really just linear algebra. – Ziggy Aug 10 '15 at 12:23
• For whatever it's worth: you can format the symbol as \prod^\curvearrowright to get $$\prod^\curvearrowright$$ not sure how to add the indices in there though – Omnomnomnom Aug 10 '15 at 12:41
• @Omnomnomnom \overset{\curvearrowright}{\prod^{k}} produces $$\overset{\curvearrowright}{\prod^{k}}$$ – wltrup Aug 10 '15 at 13:11

Matrix multiplication is not commutative, so (maybe) that symbol means that you are considering the product adding the next matrix to the right: $$X_1 X_2 \cdots X_k.$$ Anyway, I am just guessing, I've never seen it before!

• You mean "Matrix multiplication is not commutative, ..." :) – wltrup Aug 10 '15 at 13:00
• Sure! Thanks ;) – Paglia Aug 10 '15 at 13:05

The arrow notation removes ambiguity when certain procedures are represented by successive matrix multiplication.

For example, we can find $LU$ decomposition of the matrix $A$ by running Gaussian elimination on $A$. The reduced matrix is the upper triangular matrix $U$, where the reduction is represented by left multiplication of transformation matrices $E_i$ for $i \in\{1,\ldots, n-1\}$,

$$U = \left(\overset{\curvearrowleft}{\prod_{i=1}^{n-1}} E_i\right) A = E_{n-1}\cdots E_1 A.$$

Further, it can be shown that

$$L = \overset{\curvearrowright}{\prod_{i=1}^{n-1}} E_i^{-1} = E_1^{-1}\cdots E_{n-1}^{-1}$$

is a lower triangular matrix. Whence we obtain $A = LU$.

Not great for exposition, but fine as short hand.

The picture suggests that these are perhaps just two lines, which give $$\curvearrowright \prod_{i=1}^k X_i=Y$$ if you write it in one line. Then it would mean that the equation $\prod_{i=1}^k X_i=X_1X_2\cdots X_k=Y$ follows from another relation in the text before (i.e., $\cal{X} \curvearrowright \cal{Y})$.