As for "column-majorness" of matrix algebra, it seems to be natural (for me) because it is common to write a vector in column form. But of course, one could start with a row vector as the basic building block, so it looks like a matter of convention...
The reason for this choice might ultimately be traced back to the greater ratio of right-handed people (just my guess!). As English is written from left to right so as to be convenient for right-handed people with pen (again a guess!!), old mathematician in German, France, and US (where people often write from left to right) might have preferred to use right eigenvectors A x = lambda x rather than left eigenvectors x A = lambda x, in such a way that sentences begin with capital letters (Hello rather than helloW). Also, people may prefer reading tabular data from left to right because it is close to reading a sentence in a book. So the true question is why there are more right-handed people :) And because there are countries where people write from top to bottom or right to left, it would have been interesting if matrix algebra originated from such countries...
