In This lecture SVD Lecture @20:58 when the lecturer is writing down the transformation generalization:
Why is the vector matrix a square matrix? (n x n notation) What about if I have 3 vectors in a 2-dim system? wouldn't it be a 2x3 matrix?
Is it because we're 'representing' "n-spheres" in orthonormal systems? (requiring a n x n vector matrix to be defined?)
(My Linear Algebra is rusty)
Note that the vectors $\vec u_1,\dots, \vec u_m$ and $\vec u_1, \dots, \vec u_n$ are supposed to be orthonormal bases of $\Bbb R^m$ and $\Bbb R^n$ respectively. A basis for $\Bbb R^n$ (consisting of vectors on $n$ components) must contain $n$ vectors, which means that the resulting matrices will be square.