# The names of two unfamiliar operations

I am currently researching a QR-based root finding algorithm encountered two operations that I don't understand. I'd love to look them up, but I can't find the names of these operations/notations.

The first was an unfamiliar usage of a colons and commas in a possibly set related way - (1 : 2, 1). The term seemed to be expressed as the coefficient to another term (I.E. F(1 : 2, 1)). I later encountered a similar term (1 : 2, 2 : 3) that also seemed represented as a coefficient to a variable.

NOTE: After running a few more searches and rereading the article I encountered this on, I realized this notation may be matrix related rather than set related. F is actually a unitary matrix used to generate another matrix, it seems. I thought these might be matrix indices, but the typical notation for this used in the article is two comma separated subscript numbers.

Specifically, I encountered this in equation (21) under Theorem 2.8 of the paper A fast implicit QR eigenvalue algorithm for companion matrices. The equation describes a vector hk as the result of a formula Fk (1 : 2, 1) and a matrix Bk + 1 as the result of an equation Fk (1 : 2, 2 : 3).

The second is the one I had trouble running a search on. It seemed to be two terms, one over top of the other, in parenthesis as if functioning as a coefficient. I came up with the term "binomial coefficient" at some point, but I believe this might not be correct.

I'd settle for just the names of these two operations/notations, but if anyone is willing to explain what exactly they mean, it'd be appreciated. Thank you for taking the time to read this question. I apologize if this is not a suitable question for this forum or the details are inadequate.

• I'm not sure why did you decide that [set-theory] was fitting. Commented Mar 12, 2012 at 7:46
• I assumed the (1 : 2, 1) was part of set notation. I apologize for this mistake. Commented Mar 12, 2012 at 22:39
• Please provide some context. Where did you see F(1:2,1) and (1:2,2:3)? Commented Mar 13, 2012 at 0:07
• If you can quote some context from the source where you found this notation, or better yet give a reference or link to the source itself, that would probably help. It's also quite possible that this is some non-standard notation invented by that author; did you check carefully whether the author defines it previously? Commented Mar 13, 2012 at 0:47
• I have updated the question to include the information requested. I'm sorry that I had not previously included this. Commented Mar 13, 2012 at 1:00

## 2 Answers

Thanks for linking the source. It looks to me like this notation is used to refer to submatrices. So if $A$ is a matrix, $A(i:j,k:l)$ is the $(j-i+1) \times (l-k+1)$ matrix consisting of rows $i$ through $j$ and columns $k$ through $l$ of $A$. Variations are $A(i, k:l)$ to refer to the $i$th row, columns $k$ through $l$, and $A(i, :)$ to refer to the $i$th row, all columns.

Also, it appears that notation like $$\begin{pmatrix} A \\ B \end{pmatrix}$$ refers to the block matrix whose first rows come from $A$ and whose last rows come from $B$.

• Thank you (and everyone else) for your persistence in answering this question. Hopefully I can finally finish this algorithm soon. Commented Mar 13, 2012 at 1:23
• The "colon" notation here is inherited from MATLAB; in fact, books on numerical linear algebra like Golub and Van Loan's sometimes borrows idioms from MATLAB to express algorithms compactly. This paper seems to be no exception. Commented Mar 28, 2018 at 15:29

Regarding your second one, I'm not sure what kind of search you tried: did you try Wikipedia for "binomial coefficient"?

The definition is $${n\choose{k} }=\frac{n!}{k!(n-k)!},$$ usually refered in English as "$n$ choose $k$". They appear very naturally in combinatorics, but they get their name from the binomial formula $$(a+b)^n=\sum_{k=0}^n{n\choose k}a^kb^{n-k}$$

• Thank you for your answer. It seems I made a misconception about what exactly a binomial coefficient was. Commented Mar 12, 2012 at 3:54