Why are certain matrices referred to as n x n?

Yes, "rectangular" matrices represent linear transformations from R^m to Rⁿ. The problem is that linear transformations from spaces of different dimensions can't be 1 to 1 or onto- that is, they don't have inverses. It is basically the same problem as solving m equations in n unknowns. If m< n, there are not enough equations and many different solutions. If n< m, there are too many equations- in general there is no solution. It is only n equations in n unknowns that have solutions.

 

in math n is usually the first symbol used to define the mth term... thus u see M_nxn rather than M_mxm. Its like
[] "i" is usually the first counter followed by j,k,l
[] "a" is coefficient followed by b,c,d
[] "x" for spatial quantities.
[] F for functions G,H
[] theta for angles
etc. who designed them i don't know but its the way you usually learn.

 

in other words, there is no reason at all of any mathematical ,importance, you can say m by m all you want, there is no difference. indsed in logic a bound variable also known as a "dummy" variable, it could be phrased, "& by & matrix, where '&' is the number of rows and columns".

i.e. the same is expressed as well by " [ ] by [ ] matrix where [ ] = the number of rows and columns," and you can put any symbol at all in all three brackets [ ]. but people have limited imaginations, and many students are confused by creativity, so we try to keep it simple. obviously in languages written in other alphabets you are less likely to find "n by n".

i'll try to remember to look in "vectornye prostranstvye" a russian text i have written in the cyrillic alphabet.

 

actually in chinese they tend to go towards the english notations. these days. I want to use the chinese alphabet though...cuz using the roman and greek has limitations when you cross fields.