[Rayhem]

Eigen has one of if not the best linear algebra APIs I've ever seen. In particular, vectors are column vectors by default and you never need to touch row vectors (if I can editorialize, row vectors shouldn't exist at all), and vectors are not simply "n-by-1" matrices -- they're true vectors.

[vbarrielle]

But they are nx1 matrices in Eigen, you can call the rows() and cols() methods to check that. However, the nice part is that, since Eigen knows the number of columns at compile time, it enables vector methods such as size() in addition, which lets you handle them as true vector.

But tge nice part is that these vectors can still interact well with parts of the code that expect a matrix.

[mereel]

Could you elaborate a bit more about the difference between Nx1 matrices and "true vectors"?

[the_svd_doctor]

One has shape (N, 1) and one has shape (N). Always having a trailing 1 is a pain in many settings. Matlab has that “trailing 1” all the time and it’s a PITA.

[patrick451]

Why do you say it's a pain? After working with numpy for the last two years, that's the part I miss most about matlab.

[Rayhem]

There are many reasons the "-by-1" doesn't make sense.

* If a vector is an n-by-1 array (a "matrix"), why not an n-by-1-by-1-by-1-by-1?

* What is a row vector in something like a function space? Is sin(t) a row or column vector?

* You _can_ think of a matrix as acting "to the right (on a column vector)" and "to the left (on a row vector)", but this disrupts the notion of a matrix as a linear function acting on a thing (since it can act on two things). It's much more consistent to define a matrix acting on a vector, m(x), independent of "direction" and then define the transformations that let you go in the opposite direction (take the transpose of m which is a whole new matrix).

* Are physical quantities like velocity a row or column vector? Convention dictates they're column vectors, so in what scenario would it become a row vector? If I have a velocity, v1, that's a row vector and another, v2, that's a column vector, is it appropriate to calculate the projection of v1 onto v2? Should the projection operation be "aware" of the orientations? If all vectors are the same ("column vectors"), this isn't a problem -- you can always define the projection operation without special cases.

* In matlab specifically, if you have an n-by-1 array `a` and a 1-by-n array `b` with the same elements, then `a(i) == b(i)` but `a` and `b` are not the same thing.

[gugagore]

If you want a great video on the topic, see https://youtu.be/C2RO34b_oPM

Or equivalently https://jiahao.github.io/talk/2017-06-juliacon/

There are basically two self-consistent choices for the design of linear algebra libraries.