# Is there a shorthand notation that can replace an expression with $0$ if the expression is undefined?

I'm trying to write an expression for a certain operation on a matrix (namely, adding a row full of zeros) that I can use as part of a big ugly self-contained definition. Using the implicit $$\mathbf{X}_{i,j} = f(i,j)$$ notation, this is what I had, originally...

$$\left[\mathbf{X}_{i,j}\right]_{m\times n}\mapsto\left[\mathbf{X}_{i,j}\right]_{(m+1)\times n}$$

... until I noticed that when $$i=m+1$$, $$\mathbf{X}_{i,j}$$ does not have a defined value, and isn't necessarily zero by default. But I like to cover my bases. So, to make sure every entry is defined, I did this.

$$\left[\mathbf{X}_{i,j}\right]_{m\times n}\mapsto\left[\begin{cases}\mathbf{X}_{i,j}&\text{if }i\leq m\\0&\text{otherwise}\end{cases}\right]_{(m+1)\times n}$$

Now, since in my case, I'm actually adding and subtracting other things from each entry (including the ones in the new row), I find this in-expression piecewise notation to be quite ugly. I realise I could just invent some notation like putting a tilde over the $$\mathbf{X}$$ and use the piecewise expression outside of the main equation to define it, but I like to put these things in one big expression. So, what I want to know is this:

Is there a reasonably common, clean notation that could help me represent "$$\mathbf{X}_{i,j}$$, but $$0$$ if it's undefined"? If not, is there a neater expression I could use in the place of that piecewise one? Or perhaps just some simple notation for matrices that would be easily understood as "$$\mathbf{X}$$, with an additional row of zeros"?

## 1 Answer

Or perhaps just some simple notation for matrices that would be easily understood as "$$\mathbf X$$, with an additional row of zeros"?

I would suggest a block matrix:

$$\mathbf X \mapsto \begin{bmatrix} \mathbf X \\ \mathbf 0_{1\times n} \end{bmatrix}$$

In situations that are too complicated for using block matrices as a shortcut, the Iverson bracket would provide a fairly compact notation: $$(X_{ij})_{m\times n} \mapsto \bigl([i\le m]X_{ij}\bigr)_{(m+1)\times n}$$ It is a common convention that the Iverson bracket produces a "strong zero" that overrides the undefinedness of whatever it's multiplied with. On the other hand, the Iverson bracket itself is not in so common use that you can just start using it without first explaining to your reader that that's what you're doing.

• Ah, I'd somehow missed that convention on the Wikipedia article. Does that same convention apply to similar notations like the Kronecker delta? – bjshnog Jul 7 at 11:26
• @bjshnog: I'd expect it would probably be understood there too. – Henning Makholm Jul 7 at 11:49