I am an undergraduate applied mathematics student. If you are a more experienced mathematician, perhaps working in academia or whatnot, I'd like to hear your opinion on this. When writing mathematics by hand and (usually) when typesetting mathematical language, I like to denote a vector with an underline, like this: $$\underline{u}$$
Unless I'm doing physics, in which case I use an arrow: $$\vec{u}$$ However, though I typically use boldface to denote a matrix when typesetting mathematical language, this is clearly impossible when writing by hand, so I like to use a double underline, like this:$$\underline{\underline{A}}$$ Does anyone else do this? To me both notations do a good job of encapsulating their dimensional properties, as a vector is a one dimensional tensor (hence one underline) and a matrix is a two dimensional tensor (hence two underlines). Should I continue using this notation or is it unclear?
The double underline to indicate a matrix is clear and unambiguous. It is also practical for handwritten notes or for writing on a whiteboard or blackboard However, it is not conventional. Moreover, in typset documents, it is typographically both extravagant (in paper documents) and messy.
A better solution in typeset material is to avoid capital letters in denoting nonmatrix quantities, and to reserve them for matrices. If that isn't practical, use bold italic caps for matrices $\pmb A,\pmb B,...$. Bold roman $\mathbf a,\mathbf b,...\mathbf A,\mathbf B,...$ is best avoided for vectors and matrices, as it is in awkward contrast to the italic used for scalar variables among which these symbols occur. Also, roman type is best reserved for nonvariables, such as standard operators, functions and constants. In any case, use bold italic lowercase for vectors $\pmb a,\pmb b,...$.