Thread: Domain of a linear operator

Does anyone know how to find the domain of a matrix?

Ex:

B=[1,0,1;5,4,9;2,4,6]

I've tried to find a method but I don't find much on the subject.

Thanks!

The domain of an $\displaystyle m\times n$ matrix is the set of all $\displaystyle n\times p$ matrices, for all positive integers $\displaystyle p$. The only requirement is that the matrix multiplication be defined. The condition I've listed is the requirement necessary.

My apologies, but would you mind elaborating a little on that? If I have a 2x3 matrix I understand the domain is 3 x p but I don't think that answers the question entirely.

Thanks for the help!

Well, there's not a whole lot to elaborate on. You're considering the linear operator consisting of matrix multiplication on the left (that is, the operator matrix is on the left of its operand and multiplies it using regular matrix multiplication). The natural domain of a function or operator is the set of all things that you can "plug in" to the operator. For these operators, the domain is what I've described above.

Is there some more specific portion of this for which you'd like more info?

Well my specific question is what exactly does the answer look like if I'm asked to give the domain of the matrix given in the OP? I do better with examples so I apologize if that's what you've tried to describe in words.

A domain is a set, so I would use set builder notation. So, you've got something like

$\displaystyle \displaystyle \mathcal{D}(B)=\{A:A\in \mathbb{R}^{n\times p}\}.$